Orthogonality and Structured Simplification: How Linear Algebra Separates System Structure

Last Updated June 28, 2026

Orthogonality and structured simplification explain how linear algebra separates systems into directions that do not interfere with one another. When vectors are orthogonal, their relationship becomes easier to interpret, compute, approximate, and validate. Orthogonality turns complicated structure into separated components, residuals, basis directions, projections, decompositions, and stable numerical workflows.

This article introduces orthogonality as a core organizing principle in linear algebra for systems modeling. It connects dot products, perpendicular directions, orthogonal complements, orthogonal sets, orthonormal bases, projection geometry, residual diagnostics, least-squares structure, Gram-Schmidt orthogonalization, QR decomposition, orthogonal matrices, numerical stability, dimensional simplification, and responsible interpretation.

The central modeling question is not only “Are these vectors perpendicular?” It is “What system structure becomes simpler when directions are separated, independent under the chosen geometry, or decomposed into orthogonal components?”

Orthogonality is one of the most useful ideas in linear algebra because it gives a rigorous meaning to separation. Two directions can be independent in the algebraic sense without being perpendicular. Orthogonality adds geometry: one direction contributes nothing along another under the chosen inner product. This makes projection, decomposition, least squares, residual analysis, and numerical algorithms much easier to interpret.

For systems modeling, orthogonality is a tool for structured simplification. It helps separate modeled structure from residual structure, signal from noise, dominant patterns from minor variation, independent modes from coupled behavior, and stable computational directions from unstable ones. But orthogonality is not automatic independence in the real world. It depends on the geometry chosen by the model.

Why Orthogonality Matters

Orthogonality matters because it gives linear algebra a disciplined way to separate structure. When directions are orthogonal, measuring one direction does not pick up anything from the other. In geometric terms, the directions are perpendicular. In modeling terms, the components can be treated as separated under the model’s chosen geometry.

This separation makes many workflows easier. Projection becomes clean. Residuals become interpretable. Least-squares fitting becomes geometric. Orthogonal bases simplify coordinate calculations. Orthogonal matrices preserve length. QR decomposition supports stable computation. Principal components and singular vectors use orthogonality to organize high-dimensional structure.

Orthogonality is therefore a simplification principle. It does not remove complexity by pretending the system is simple. It separates the complexity into structured parts.

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Dot Products and Perpendicular Directions

The dot product measures directional alignment. For vectors \(\mathbf{u}\) and \(\mathbf{v}\), the dot product combines magnitude and angle:

If the dot product is positive, the vectors point partly in the same direction. If it is negative, they point partly in opposite directions. If it is zero, they are orthogonal under the standard Euclidean geometry.

The dot product turns geometric alignment into a computable diagnostic.

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Orthogonal Vectors

Two nonzero vectors are orthogonal when their dot product is zero. This means neither vector has any component in the direction of the other. Orthogonality is stronger than linear independence for two nonzero vectors: if two nonzero vectors are orthogonal, they must be linearly independent.

This matters in modeling because orthogonal directions create clean separation. A state can be decomposed into components along different directions without overlap. A residual can be checked against model directions. A basis can be designed to avoid coordinate interference.

Orthogonality gives independence a geometric meaning, but it should always be interpreted relative to a chosen inner product.

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Orthogonal Sets

A set of vectors is orthogonal if every pair of distinct vectors in the set is orthogonal. If all vectors in the set are nonzero, then an orthogonal set is automatically linearly independent.

Orthogonal sets simplify coordinate calculations because each direction can be measured separately. There is no overlap among basis directions under the chosen geometry.

Orthogonal sets are a bridge from geometric separation to stable representation.

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Orthonormal Bases

An orthonormal basis is an orthogonal basis whose vectors all have unit length. If \(\{\mathbf{q}_1,\ldots,\mathbf{q}_n\}\) is an orthonormal basis, then:

In an orthonormal basis, coordinates are especially simple. The coordinate of \(\mathbf{x}\) along \(\mathbf{q}_i\) is just the dot product \(\mathbf{x}\cdot\mathbf{q}_i\).

Orthonormal bases are central to scientific computing because they reduce numerical instability. They appear in QR decomposition, least squares, singular value decomposition, principal component analysis, Fourier methods, and many stable algorithms.

Orthonormal bases provide one of the cleanest forms of structured simplification in linear algebra.

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Orthogonal Complements

The orthogonal complement of a subspace \(S\) is the set of all vectors orthogonal to every vector in \(S\). It is written \(S^\perp\).

Orthogonal complements explain the geometry of residuals. If a vector is projected onto a subspace, the residual lies in the orthogonal complement of that subspace.

Orthogonal complements make residual structure mathematically visible.

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Orthogonal Decomposition

Orthogonal decomposition expresses a vector as a sum of components that lie in orthogonal subspaces. In the simplest case, a vector is split into one component inside a subspace and one component inside its orthogonal complement.

This is one of the cleanest forms of structured simplification. It allows a modeler to say: this part of the state is captured by the representation, and this part is not. That distinction is fundamental to residual analysis, least squares, signal processing, dimensional reduction, and scientific model validation.

Orthogonal decomposition supports both calculation and interpretation because the parts are separated rather than tangled.

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Projection and Residual Structure

Projection and orthogonality are inseparable. Orthogonal projection finds the closest point in a subspace because the residual is perpendicular to the subspace. This is the geometric basis of approximation.

When a projection matrix \(P\) is orthogonal, \(P\mathbf{x}\) gives the retained component and \((I-P)\mathbf{x}\) gives the residual component.

This structure matters because residuals are not merely calculation artifacts. They are evidence. A residual can mean noise, measurement error, excluded mechanism, model misspecification, structural bias, or a real pattern that the chosen subspace cannot represent.

Orthogonality gives projection its explanatory power.

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Least Squares and Orthogonality

Least squares is an orthogonality condition. When \(A\mathbf{x}=\mathbf{b}\) is inconsistent, the least-squares solution \(\widehat{\mathbf{x}}\) produces a fitted vector \(A\widehat{\mathbf{x}}\) such that the residual is orthogonal to the column space of \(A\).

This condition says that no model direction can explain what remains in the residual. If a residual still had a component along a model column, the fit could be improved by moving in that direction.

Least squares is therefore not just optimization. It is orthogonal projection onto a model subspace.

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Gram-Schmidt Orthogonalization

Gram-Schmidt orthogonalization converts a linearly independent set of vectors into an orthogonal or orthonormal set spanning the same subspace. It does this by subtracting off components already explained by earlier directions.

After orthogonal directions are found, they can be normalized:

Conceptually, Gram-Schmidt is a structured simplification procedure. It takes overlapping directions and separates them into non-overlapping components. Numerically, however, classical Gram-Schmidt can be unstable for nearly dependent vectors. Modified Gram-Schmidt, Householder QR, or SVD may be preferable in serious scientific computing.

Orthogonalization is useful because it turns overlap into organized separation.

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Orthogonal Matrices

A square matrix \(Q\) is orthogonal if its columns form an orthonormal basis. Equivalently:

For an orthogonal matrix, the inverse is the transpose:

Orthogonal matrices preserve Euclidean lengths and angles. They represent rotations, reflections, and coordinate changes that do not distort Euclidean geometry.

Orthogonal matrices are valuable because they change representation without amplifying numerical fragility.

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QR Decomposition

QR decomposition factors a matrix \(A\) into an orthogonal or orthonormal matrix \(Q\) and an upper triangular matrix \(R\):

QR decomposition is one of the most important computational uses of orthogonality. It supports stable least-squares solutions, rank diagnostics, regression workflows, eigenvalue algorithms, and scientific computing pipelines.

In least squares, QR avoids explicitly forming \(A^TA\), which can square the condition number and worsen numerical instability. If \(A=QR\), then solving \(A\mathbf{x}\approx\mathbf{b}\) can be reduced to solving a triangular system involving \(R\).

QR decomposition is orthogonality turned into a practical solver workflow.

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Structured Simplification

Orthogonality simplifies systems by reducing overlap. In a nonorthogonal representation, one coordinate may partially duplicate another. In an orthogonal representation, each direction contributes separately under the chosen geometry. This makes interpretation, approximation, compression, and diagnostics cleaner.

Structured simplification does not mean oversimplification. It means choosing a representation in which different pieces of the system can be examined separately. Orthogonal bases, projections, QR decomposition, PCA, SVD, and Fourier analysis all use some form of this principle.

Orthogonality provides a disciplined way to simplify without losing track of what has been separated, retained, or discarded.

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Numerical Stability and Conditioning

Orthogonality is central to numerical stability. Orthogonal transformations do not magnify errors in the Euclidean norm. This is why stable algorithms often prefer orthogonal transformations over direct inversion or normal equations.

For example, forming \(A^TA\) can worsen conditioning because the condition number may be squared. QR decomposition avoids this by using orthogonal transformations to preserve numerical structure more reliably.

Orthogonality is not only geometrically elegant. It is a practical safeguard against numerical fragility.

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Orthogonality in Systems Modeling

Orthogonality appears throughout systems modeling. In infrastructure analysis, orthogonal modes can separate vulnerability patterns. In economics, orthogonal factors can separate independent variation patterns in sectoral activity. In ecology, orthogonal gradients can separate environmental drivers. In machine learning, orthogonal features or embeddings can reduce redundancy. In scientific computing, orthogonal decompositions stabilize algorithms.

But orthogonality is model-relative. Two vectors are orthogonal under a chosen inner product, scaling, and representation. If the units or weights change, the orthogonality relation can change. This matters in policy, economics, ecology, infrastructure, public health, and institutional modeling, where variables may have different units, uncertainty levels, and normative significance.

Orthogonality can clarify system structure, but only when the chosen geometry is defensible.

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Mathematical Deepening

This section adds a more formal layer. Orthogonality connects inner product spaces, orthogonal complements, projections, direct sums, normal equations, Gram-Schmidt, QR factorization, orthogonal matrices, unitary analogues, spectral theory, PCA, SVD, Fourier analysis, Hilbert-space generalization, and stable numerical algorithms.

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Computation and Reproducible Workflows

Computational workflows for orthogonality should document input vectors or matrix columns, dot products, orthogonality tolerances, orthonormalization method, \(Q^TQ-I\) error, projection residuals, QR outputs, rank warnings, condition warnings, and interpretation notes. Orthogonality should not be assumed from visual diagrams or rounded output alone.

The companion repository treats orthogonality as auditable systems modeling. Python, R, Julia, SQL, Haskell, C, C++, Fortran, Rust, Go, notebooks, schemas, generated outputs, Canvas artifacts, advanced reports, and calculators each support a different layer of reproducible orthogonal-structure analysis.

For this article, the computational examples focus on dot-product checks, orthonormal basis construction, projection residuals, QR-style diagnostics, orthogonality error, and model-governance records.

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Python Workflow: Orthogonality Audit

The Python workflow below checks orthogonality among vectors, constructs normalized directions, computes projection and residual structure, and records numerical diagnostics.

from __future__ import annotations

from dataclasses import asdict, dataclass
from pathlib import Path
import csv
import json
import math


Vector = list[float]
Matrix = list[list[float]]


@dataclass(frozen=True)
class OrthogonalityAudit:
    system_name: str
    vector_a: str
    vector_b: str
    dot_product: float
    orthogonal_under_tolerance: bool
    unit_a: str
    unit_b: str
    projection_of_a_onto_b: str
    residual_vector: str
    residual_norm: float
    orthonormality_error: float
    interpretation_warning: str


def dot(a: Vector, b: Vector) -> float:
    return sum(x * y for x, y in zip(a, b))


def norm2(v: Vector) -> float:
    return math.sqrt(dot(v, v))


def scale(v: Vector, c: float) -> Vector:
    return [c * x for x in v]


def subtract(a: Vector, b: Vector) -> Vector:
    return [x - y for x, y in zip(a, b)]


def normalize(v: Vector) -> Vector:
    n = norm2(v)
    if n <= 1e-12:
        raise ValueError("cannot normalize near-zero vector")
    return [x / n for x in v]


def projection(a: Vector, b: Vector) -> Vector:
    denom = dot(b, b)
    if denom <= 1e-12:
        raise ValueError("cannot project onto near-zero vector")
    return scale(b, dot(a, b) / denom)


def vector_to_string(v: Vector) -> str:
    return ",".join(f"{x:.6f}" for x in v)


def build_audit() -> OrthogonalityAudit:
    a = [3.0, 1.0, 2.0]
    b = [1.0, -1.0, -1.0]

    dot_ab = dot(a, b)
    tol = 1e-10
    unit_a = normalize(a)
    unit_b = normalize(b)

    proj_ab = projection(a, b)
    residual = subtract(a, proj_ab)

    # Two-column orthonormality check for normalized a and b.
    q_dot = dot(unit_a, unit_b)
    orthonormality_error = abs(q_dot)

    return OrthogonalityAudit(
        system_name="three_component_orthogonality_audit",
        vector_a=vector_to_string(a),
        vector_b=vector_to_string(b),
        dot_product=round(dot_ab, 12),
        orthogonal_under_tolerance=abs(dot_ab) <= tol,
        unit_a=vector_to_string(unit_a),
        unit_b=vector_to_string(unit_b),
        projection_of_a_onto_b=vector_to_string(proj_ab),
        residual_vector=vector_to_string(residual),
        residual_norm=round(norm2(residual), 12),
        orthonormality_error=round(orthonormality_error, 12),
        interpretation_warning=(
            "Orthogonality depends on the chosen inner product, scaling, units, and tolerance. "
            "A residual that is orthogonal to model directions may still contain important excluded structure."
        ),
    )


def write_outputs(output_dir: Path) -> None:
    (output_dir / "tables").mkdir(parents=True, exist_ok=True)
    (output_dir / "json").mkdir(parents=True, exist_ok=True)

    audit = build_audit()
    row = asdict(audit)

    with (output_dir / "tables" / "orthogonality_audit.csv").open(
        "w", newline="", encoding="utf-8"
    ) as handle:
        writer = csv.DictWriter(handle, fieldnames=list(row.keys()))
        writer.writeheader()
        writer.writerow(row)

    (output_dir / "json" / "orthogonality_audit.json").write_text(
        json.dumps(row, indent=2, sort_keys=True),
        encoding="utf-8",
    )


if __name__ == "__main__":
    write_outputs(Path("outputs"))
    print("Orthogonality audit complete.")

This workflow treats orthogonality as an auditable modeling claim. It records dot products, tolerance decisions, normalized directions, projection structure, residual magnitude, orthonormality error, and interpretation warnings.

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R Workflow: Orthogonal Structure Diagnostics

R can support orthogonality diagnostics by checking dot products, projections, residuals, normalized directions, and QR decomposition outputs.

a <- c(3, 1, 2)
b <- c(1, -1, -1)

dot_ab <- sum(a * b)
tol <- 1e-10
orthogonal_under_tolerance <- abs(dot_ab) <= tol

norm2 <- function(x) sqrt(sum(x^2))
unit_a <- a / norm2(a)
unit_b <- b / norm2(b)

projection_of_a_onto_b <- (sum(a * b) / sum(b * b)) * b
residual <- a - projection_of_a_onto_b
residual_norm <- norm2(residual)

Q <- cbind(unit_a, unit_b)
orthonormality_error <- sqrt(sum((t(Q) %*% Q - diag(2))^2))

audit_record <- data.frame(
  system_name = "three_component_orthogonality_audit",
  vector_a = paste(round(a, 6), collapse = ","),
  vector_b = paste(round(b, 6), collapse = ","),
  dot_product = dot_ab,
  orthogonal_under_tolerance = orthogonal_under_tolerance,
  unit_a = paste(round(unit_a, 6), collapse = ","),
  unit_b = paste(round(unit_b, 6), collapse = ","),
  projection_of_a_onto_b = paste(round(projection_of_a_onto_b, 6), collapse = ","),
  residual_vector = paste(round(residual, 6), collapse = ","),
  residual_norm = residual_norm,
  orthonormality_error = orthonormality_error,
  interpretation_warning = paste(
    "Orthogonality depends on the chosen inner product, scaling, units, and tolerance.",
    "Orthogonal residuals may still contain important excluded structure."
  )
)

dir.create("outputs/tables", recursive = TRUE, showWarnings = FALSE)

write.csv(
  audit_record,
  "outputs/tables/r_orthogonality_audit.csv",
  row.names = FALSE
)

print(audit_record)

This R workflow is useful when orthogonal structure supports least squares, residual diagnostics, QR workflows, model simplification, or high-dimensional data review.

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Haskell Workflow: Typed Orthogonality Records

Haskell can represent orthogonality diagnostics as a typed record with fields for dot products, tolerance decisions, projection results, residual norms, and interpretation warnings.

module Main where

data OrthogonalityAudit = OrthogonalityAudit
  { systemName :: String
  , vectorA :: String
  , vectorB :: String
  , dotProduct :: Double
  , orthogonalUnderTolerance :: Bool
  , unitA :: String
  , unitB :: String
  , projectionOfAOntoB :: String
  , residualVector :: String
  , residualNorm :: Double
  , orthonormalityError :: Double
  , interpretationWarning :: String
  } deriving (Show)

buildAudit :: OrthogonalityAudit
buildAudit =
  OrthogonalityAudit
    "three_component_orthogonality_audit"
    "3.000000,1.000000,2.000000"
    "1.000000,-1.000000,-1.000000"
    0.0
    True
    "0.801784,0.267261,0.534522"
    "0.577350,-0.577350,-0.577350"
    "0.000000,0.000000,0.000000"
    "3.000000,1.000000,2.000000"
    3.741657
    0.0
    "Orthogonality depends on geometry, scaling, units, and tolerance; residuals require substantive interpretation."

main :: IO ()
main =
  print buildAudit

The typed workflow keeps orthogonality checks attached to the modeling interpretation, rather than reducing them to anonymous dot-product calculations.

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SQL Workflow: Orthogonality Assumption Registry

SQL can document orthogonality assumptions when geometric simplification supports dashboards, scientific-computing pipelines, residual reviews, model documentation, or institutional decision workflows.

CREATE TABLE orthogonality_assumption_registry (
    assumption_key TEXT PRIMARY KEY,
    assumption_name TEXT NOT NULL,
    mathematical_role TEXT NOT NULL,
    systems_modeling_role TEXT NOT NULL,
    review_warning TEXT NOT NULL
);

INSERT INTO orthogonality_assumption_registry VALUES
(
  'inner_product',
  'Inner product',
  'Defines alignment, angle, length, and orthogonality.',
  'Determines what it means for system components to be separated.',
  'Changing weights, units, or scaling can change orthogonality.'
);

INSERT INTO orthogonality_assumption_registry VALUES
(
  'dot_product_zero',
  'Zero dot product',
  'Defines perpendicularity under the standard Euclidean inner product.',
  'Indicates no overlap between directions under the chosen geometry.',
  'Zero dot product is not automatically real-world independence.'
);

INSERT INTO orthogonality_assumption_registry VALUES
(
  'orthogonal_set',
  'Orthogonal set',
  'A set of vectors with pairwise zero inner products.',
  'Provides separated directions for representation or decomposition.',
  'Vectors must be nonzero to form a basis.'
);

INSERT INTO orthogonality_assumption_registry VALUES
(
  'orthonormal_basis',
  'Orthonormal basis',
  'A basis of mutually orthogonal unit vectors.',
  'Supports simple coordinates, stable decomposition, and clean projection.',
  'Coordinates may be abstract even when numerically convenient.'
);

INSERT INTO orthogonality_assumption_registry VALUES
(
  'orthogonal_projection',
  'Orthogonal projection',
  'Separates vectors into subspace and orthogonal residual components.',
  'Distinguishes modeled structure from residual structure.',
  'Residuals may contain important excluded signal or bias.'
);

INSERT INTO orthogonality_assumption_registry VALUES
(
  'qr_decomposition',
  'QR decomposition',
  'Factors a matrix into orthonormal directions and triangular coefficients.',
  'Supports stable least-squares and structured simplification workflows.',
  'Rank and conditioning should still be reviewed.'
);

SELECT
    assumption_name,
    mathematical_role,
    systems_modeling_role,
    review_warning
FROM orthogonality_assumption_registry
ORDER BY assumption_key;

This registry keeps orthogonality tied to inner products, zero-overlap claims, orthonormal bases, projection geometry, QR workflows, and responsible interpretation.

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GitHub Repository

The companion repository for this article is designed as a reproducible mathematical-modeling workspace. It supports orthogonality audits, dot-product diagnostics, orthogonal-set checks, projection residual reports, QR-style decomposition records, SQL governance tables, generated outputs, advanced mathematical audit reports, and reusable calculator scripts.

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Interpretive Limits and Responsible Use

Orthogonality is powerful because it separates structure cleanly. It is limited because the separation depends on the chosen geometry. If variables have different units, uncertainty levels, scales, or institutional meanings, the standard dot product may not be the right measure of overlap. A weighted inner product or domain-specific metric may be more appropriate.

Orthogonal components are not automatically causal components. Principal components, orthogonal factors, or QR directions may simplify computation without corresponding to real mechanisms. A residual orthogonal to model columns may still contain important excluded structure, bias, measurement error, nonlinear behavior, or domain-relevant signal. Orthogonality can clarify a model, but it can also make transformed coordinates harder to explain.

Responsible use requires documenting the inner product, scaling, units, tolerance, orthogonalization method, residual interpretation, numerical diagnostics, rank and conditioning checks, and whether orthogonal structure is being used for explanation, approximation, compression, prediction, simulation, or decision support.

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Related Articles

• What Is Linear Algebra for Systems Modeling?
• Scalars, Vectors, and System States
• Vector Spaces and System Representation
• Span, Linear Independence, and Basis
• Dimension and the Structure of Solution Spaces
• Matrices and the Organization of Multivariable Systems
• Matrix Arithmetic and the Logic of Combination
• Systems of Linear Equations
• Gaussian Elimination and Row Reduction
• Pivot Structure and Solvability
• Rank, Nullity, and Structural Dependence
• Determinants and Invertibility
• Inverse Matrices and Structural Recovery
• Overdetermined Systems and Least Squares Thinking
• Linear Transformations and Model Behavior
• Matrix Multiplication and Interaction Effects
• Change of Basis and Alternative Representations
• Projections, Reflections, and Geometric Interpretation
• Linear Algebra for Systems Modeling
• Mathematical Modeling
• Systems Modeling
• Scientific Computing for Systems Modeling

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Further Reading

• Axler, S. (2024) Linear Algebra Done Right. 4th edn. Cham: Springer. Available at: https://linear.axler.net/.
• Boyd, S. and Vandenberghe, L. (2018) Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares. Cambridge: Cambridge University Press. Available at: https://vmls-book.stanford.edu/.
• Boyd, S. and Vandenberghe, L. (2018) Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares. PDF edition. Stanford University and UCLA. Available at: https://web.stanford.edu/~boyd/vmls/vmls.pdf.
• Georgia Institute of Technology (n.d.) Interactive Linear Algebra. Georgia Institute of Technology. Available at: https://textbooks.math.gatech.edu/ila/.
• Golub, G.H. and Van Loan, C.F. (2013) Matrix Computations. 4th edn. Baltimore, MD: Johns Hopkins University Press. Available at: https://www.press.jhu.edu/books/title/10678/matrix-computations.
• Hefferon, J. (n.d.) Linear Algebra. Saint Michael’s College. Available at: https://joshua.smcvt.edu/linearalgebra/.
• Higham, N.J. (2002) Accuracy and Stability of Numerical Algorithms. 2nd edn. Philadelphia, PA: Society for Industrial and Applied Mathematics. Available at: https://epubs.siam.org/doi/book/10.1137/1.9780898718027.
• Horn, R.A. and Johnson, C.R. (2013) Matrix Analysis. 2nd edn. Cambridge: Cambridge University Press. Available at: https://www.cambridge.org/highereducation/books/matrix-analysis/FDA3627DC2B9F5C3DF2FD8C3CC136B48.
• Julia Documentation (n.d.) Linear Algebra — Julia Standard Library. Julia Documentation. Available at: https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/.
• Massachusetts Institute of Technology OpenCourseWare (2011) Linear Algebra. Cambridge, MA: MIT OpenCourseWare. Available at: https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/.
• Massachusetts Institute of Technology OpenCourseWare (2018) Matrix Methods in Data Analysis, Signal Processing, and Machine Learning. Cambridge, MA: MIT OpenCourseWare. Available at: https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/.
• Meyer, C.D. (2023) Matrix Analysis and Applied Linear Algebra. 2nd edn. Philadelphia, PA: Society for Industrial and Applied Mathematics. Available at: https://epubs.siam.org/doi/book/10.1137/1.9781611977448.
• MIT Mathematics (2023) Introduction to Linear Algebra, Sixth Edition. Cambridge, MA: Massachusetts Institute of Technology. Available at: https://math.mit.edu/~gs/linearalgebra/ila6/indexila6.html.
• Netlib (n.d.) BLAS — Basic Linear Algebra Subprograms. Netlib Repository. Available at: https://www.netlib.org/blas/.
• Netlib (n.d.) LAPACK — Linear Algebra PACKage. Netlib Repository. Available at: https://www.netlib.org/lapack/.
• Netlib and SIAM (1999) LAPACK Users’ Guide. 3rd edn. Philadelphia, PA: Society for Industrial and Applied Mathematics. Available at: https://www.netlib.org/lapack/lug/.
• NumPy Developers (n.d.) Linear Algebra: numpy.linalg. NumPy Documentation. Available at: https://numpy.org/doc/stable/reference/routines.linalg.html.
• SciPy Developers (n.d.) Linear Algebra: scipy.linalg. SciPy Documentation. Available at: https://docs.scipy.org/doc/scipy/reference/linalg.html.
• Strang, G. (2016) Introduction to Linear Algebra. Wellesley, MA: Wellesley-Cambridge Press. Available at: https://math.mit.edu/~gs/linearalgebra/.
• Treil, S. (2017) Linear Algebra Done Wrong. Providence, RI: Brown University. Available at: https://www.math.brown.edu/streil/papers/LADW/LADW.html.
• Trefethen, L.N. and Bau, D. (1997) Numerical Linear Algebra. Philadelphia, PA: Society for Industrial and Applied Mathematics. Available at: https://epubs.siam.org/doi/book/10.1137/1.9780898719574.
• van de Geijn, R. and Myers, M. (n.d.) Linear Algebra: Foundations to Frontiers. Austin, TX: The University of Texas at Austin. Available at: https://www.cs.utexas.edu/~flame/laff/laff/LAFF-2.00M.pdf.

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References

• Axler, S. (2024) Linear Algebra Done Right. 4th edn. Cham: Springer. Available at: https://linear.axler.net/.
• Boyd, S. and Vandenberghe, L. (2018) Introduction to Applied Linear Algebra: Vectors, Matrices, and Least Squares. Cambridge: Cambridge University Press. Available at: https://vmls-book.stanford.edu/.
• Georgia Institute of Technology (n.d.) Interactive Linear Algebra. Georgia Institute of Technology. Available at: https://textbooks.math.gatech.edu/ila/.
• Golub, G.H. and Van Loan, C.F. (2013) Matrix Computations. 4th edn. Baltimore, MD: Johns Hopkins University Press. Available at: https://www.press.jhu.edu/books/title/10678/matrix-computations.
• Hefferon, J. (n.d.) Linear Algebra. Saint Michael’s College. Available at: https://joshua.smcvt.edu/linearalgebra/.
• Higham, N.J. (2002) Accuracy and Stability of Numerical Algorithms. 2nd edn. Philadelphia, PA: Society for Industrial and Applied Mathematics. Available at: https://epubs.siam.org/doi/book/10.1137/1.9780898718027.
• Horn, R.A. and Johnson, C.R. (2013) Matrix Analysis. 2nd edn. Cambridge: Cambridge University Press. Available at: https://www.cambridge.org/highereducation/books/matrix-analysis/FDA3627DC2B9F5C3DF2FD8C3CC136B48.
• Massachusetts Institute of Technology OpenCourseWare (2011) Linear Algebra. Cambridge, MA: MIT OpenCourseWare. Available at: https://ocw.mit.edu/courses/18-06sc-linear-algebra-fall-2011/.
• Massachusetts Institute of Technology OpenCourseWare (2018) Matrix Methods in Data Analysis, Signal Processing, and Machine Learning. Cambridge, MA: MIT OpenCourseWare. Available at: https://ocw.mit.edu/courses/18-065-matrix-methods-in-data-analysis-signal-processing-and-machine-learning-spring-2018/.
• Meyer, C.D. (2023) Matrix Analysis and Applied Linear Algebra. 2nd edn. Philadelphia, PA: Society for Industrial and Applied Mathematics. Available at: https://epubs.siam.org/doi/book/10.1137/1.9781611977448.
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