Linear Independence in Abstract Vector Spaces

Q: How many vectors can an independent set contain?

At most the dimension of the space. In $P_4$, which has dimension five, no six polynomials can be independent; in $M_{22}$, which has dimension four, no five matrices can be. The reason is that the coefficient matrix would then have more columns than rows and could not attain full column rank.

Q: Does linear independence in a polynomial space mean the polynomials have no common roots?

No, the two notions are unrelated. The polynomials $x - 2$, $(x-2)^2$ and $(x-2)^3$ share the root $x = 2$ and are linearly independent. Conversely two polynomials with no common root can be dependent if one is a scalar multiple of the other, which is impossible only when the multiple is non-trivial.

Overview

Linear independence is the formal expression of "no redundancy". A set of vectors is independent when none of them can be reconstructed from the others, so every member contributes something the rest cannot supply. That informal statement is not directly testable, and the standard definition replaces it with one that is: independence is the absence of any non-trivial way of combining the vectors to reach the zero vector.

Because the definition mentions only scalars, vectors, addition, scalar multiplication and the zero vector, it transfers unchanged from column vectors to any vector space. The apparatus that supports it — a relation of linear dependence, the distinction between the trivial relation and a non-trivial one — also transfers, with the single hypothesis that the ambient set is a vector space. Polynomials, matrices and functions are tested with exactly the definitions written for columns.

The word carrying the weight in the definition is only. Every set admits at least one relation of linear dependence, obtained by taking all coefficients zero, and that trivial relation is never informative. Independence is the statement that no other relation exists. The same emphasis appears in the definition of a nonsingular matrix, where a homogeneous system is required to have only the trivial solution, and the parallel is not an accident: the two notions are two readings of the same computation.

That computation is the reason the abstract setting costs nothing. Writing a general relation of linear dependence and equating coordinates — coefficients for polynomials, entries for matrices — produces a homogeneous linear system in the unknown scalars. A homogeneous system is always consistent, so the only question is whether it has free variables. No free variables means the trivial solution is unique and the set is independent; one or more free variables means non-trivial relations exist and the set is dependent, and any non-trivial solution exhibits one explicitly.

Definition

Relation of Linear Dependence

RLD

Let $V$ be a vector space and $S = \{u_1, u_2, \ldots, u_n\}$ a set of vectors from $V$ . An equation of the form
$\alpha_1 u_1 + \alpha_2 u_2 + \cdots + \alpha_n u_n = \vec{0}$
is a relation of linear dependence on $S$ . If every coefficient is zero, that is $\alpha_i = 0$ for $1 \le i \le n$ , the relation is called trivial.

The trivial relation always holds, in every vector space, for every set. It carries no information and is never evidence of anything.

Linear Independence

Let $V$ be a vector space and $S = \{u_1, u_2, \ldots, u_n\}$ a set of vectors from $V$ . The set $S$ is linearly dependent if there exists a relation of linear dependence on $S$ that is not trivial. If the trivial relation is the only relation of linear dependence on $S$ , then $S$ is linearly independent.

The emphasis on only mirrors the definition of a nonsingular matrix, where a homogeneous system is required to have the trivial solution and no other.

Concepts

The evidential asymmetry

Dependence and independence are not symmetric claims. To establish dependence it suffices to exhibit one non-trivial relation and verify the arithmetic; the reader needs no knowledge of where the coefficients came from. To establish independence it is necessary to rule out every possible non-trivial relation, which cannot be done by inspection or by sampling. The asymmetry is why an independence argument always ends in a general solution of a homogeneous system, while a dependence argument can end in a single displayed identity.

Reduction to a homogeneous system

Write a general relation of linear dependence with unknown coefficients, expand it using the operations of $V$ , and equate coordinates. Since the right-hand side is the zero vector, every equation has zero on the right, so the system is homogeneous. A homogeneous system is always consistent — the trivial solution is available — and therefore the only meaningful output of the reduction is the number of free variables. If $n$ vectors are being tested and the coefficient matrix has rank $r$ , then there are $n - r$ free variables.

The rank criterion

Assembling the coordinate vectors of $u_1, \ldots, u_n$ as the columns of a matrix $B$ , the relation $\alpha_1 u_1 + \cdots + \alpha_n u_n = \vec{0}$ becomes $B\vec{\alpha} = \vec{0}$ . The set is independent exactly when $\mathcal{N}(B) = \{\vec{0}\}$ , that is, when $\operatorname{rank}(B) = n$ . This is the practical test. It also delivers an immediate corollary: in a space of dimension $d$ , any set of more than $d$ vectors is dependent, because $B$ has more columns than rows and cannot have rank $n$ .

Dependence means one vector is redundant

If a non-trivial relation exists, some coefficient $\alpha_k$ is non-zero, and dividing through by it expresses $u_k$ as a linear combination of the remaining vectors. Removing $u_k$ therefore leaves the span unchanged. This is the operational meaning of dependence: a dependent set contains at least one vector that adds nothing to what the others already reach. The converse direction is equally useful — if no vector is a combination of the others, no non-trivial relation can exist.

Independence gives unique representation

If $S$ is independent and a vector $y$ is expressed as a linear combination of $S$ in two ways, subtracting the two expressions produces a relation of linear dependence whose coefficients are the differences. Independence forces every difference to be zero, so the two expressions coincide. Representation in terms of an independent set is therefore unique, which is precisely the property that makes a basis useful and makes coordinates well defined.

Shortcuts that follow from the definition

Several consequences fall out immediately and are worth having to hand. Any set containing the zero vector is dependent, since a coefficient of one on $\vec{0}$ and zero elsewhere is a non-trivial relation. Any set containing two equal vectors is dependent. A single non-zero vector always forms an independent set. A set of two vectors is dependent exactly when one is a scalar multiple of the other — a test that does not generalise to three or more vectors, where pairwise non-proportionality is compatible with dependence.

Deciding whether a set is linearly independent

Screen for obvious dependenceIf

\vec{0}

is in the set, if two vectors coincide, or if the set has more vectors than the dimension of

V

, it is dependent. Stop.

Write a general relationSet

\alpha_1 u_1 + \cdots + \alpha_n u_n = \vec{0}

with the scalars unknown.

Expand and equate coordinatesUse the operations of

V

to collect terms, then equate coefficients or entries. Every equation has zero on the right.

Row-reduce the coefficient matrixNo augmented column is needed — the system is homogeneous and therefore always consistent.

Count free variablesIf

n - r = 0

the only solution is trivial and the set is independent. If

n - r > 0

the set is dependent.

If dependent, exhibit a witnessChoose free variables not all zero, solve for the dependent ones, and display the resulting non-trivial relation.

Equations

Relation of linear dependence

EQ-LIVS-01

α_{1} u_{1} + α_{2} u_{2} + α_{3} u_{3} + \dots + α_{n} u_{n} = \vec{0}

The right-hand side is the zero vector of $V$ , not the scalar zero. Identifying it correctly is the only translation step needed when moving between spaces.

The trivial relation

EQ-LIVS-02

0 u_{1} + 0 u_{2} + 0 u_{3} + \dots + 0 u_{n} = \vec{0}

Available for every set in every vector space. Its existence proves nothing; independence is the claim that nothing else is available.

Independence as a null space condition

EQ-LIVS-03

S linearly independent \Leftrightarrow N (B) = {\vec{0}} \Leftrightarrow rank (B) = n

Here $B$ is the matrix whose columns are the coordinate vectors of the members of $S$ . The whole test collapses to a rank computation.

Free variable count

EQ-LIVS-04

number of free variables = n - r, r = rank (B)

Zero free variables means the trivial solution is unique and the set is independent. Each free variable is one independent direction of redundancy.

Solving a non-trivial relation for a redundant vector

EQ-LIVS-05

Solving a non-trivial relation for a redundant vector

Any vector carrying a non-zero coefficient in a non-trivial relation is expressible in terms of the others, so removing it leaves the span unchanged.

Uniqueness of representation under independence

EQ-LIVS-06

\sum α_{i} u_{i} = \sum β_{i} u_{i} \Rightarrow \sum (α_{i} - β_{i}) u_{i} = \vec{0} \Rightarrow α_{i} = β_{i} for all i

Subtracting two representations produces a relation of linear dependence; independence forces every coefficient difference to vanish.

A non-trivial relation in $M_{22}$

EQ-LIVS-07

A non-trivial relation in $M_{22}$

A complete, self-contained proof of dependence: the coefficients are not all zero and the arithmetic is verifiable entry by entry.

Variable Definitions

Symbols used on this page
Symbol	Name	Meaning	Domain / type
$V$	Ambient vector space	The space in which the vectors live and combinations are formed	any vector space
$S$	Candidate set	The finite set $\{u_1, \ldots, u_n\}$ under test	finite subset of V
$n$	Set size	Number of vectors in $S$ ; equals the number of unknowns in the homogeneous system	positive integer
$α_{i}$	Coefficient	Unknown scalar attached to $u_i$ in a relation of linear dependence	complex number
$B$	Coordinate matrix	Matrix whose columns are the coordinate vectors of $u_1, \ldots, u_n$	d x n matrix
$d$	Coordinate count	Number of coordinates describing a vector of $V$ ; the number of equations produced	positive integer
$r$	Rank	Rank of the coefficient matrix $B$ ; the number of pivot columns	0 to min(d,n)
$\vec{0}$	Zero vector	The additive identity of $V$ ; the right-hand side of every relation	element of V
$M_{22}$	Matrix space	All $2 \times 2$ matrices with complex entries; dimension four	vector space

Worked Numerical Example

Problem statement

A cascaded control model stores three $2 \times 2$ coupling blocks. Establish whether the stored set is redundant, exhibit the redundancy explicitly if it exists, and then test a repaired set that replaces the offending block.

Name the vectors and the space
Work in $V = M_{22}$ with entrywise operations. The zero vector of $V$ is the $2 \times 2$ zero matrix, which is the right-hand side of every relation below.
$B_{1} = [\begin{matrix} 1 & 2 \\ - 1 & 0 \end{matrix}], B_{2} = [\begin{matrix} 0 & 1 \\ 3 & 2 \end{matrix}], B_{3} = [\begin{matrix} 2 & 3 \\ - 5 & - 2 \end{matrix}]$
Write a general relation of linear dependence
Introduce unknown scalars and expand the combination entry by entry using the operations of $M_{22}$ :
$α_{1} B_{1} + α_{2} B_{2} + α_{3} B_{3} = [\begin{matrix} α_{1} + 2 α_{3} & 2 α_{1} + α_{2} + 3 α_{3} \\ - α_{1} + 3 α_{2} - 5 α_{3} & 2 α_{2} - 2 α_{3} \end{matrix}]$
Equate entries to a homogeneous system
Setting the expanded matrix equal to the zero matrix gives four equations in three unknowns, every one with zero on the right. No augmented column is required.
${\begin{cases} α_{1} + 2 α_{3} = 0 \\ 2 α_{1} + α_{2} + 3 α_{3} = 0 \\ - α_{1} + 3 α_{2} - 5 α_{3} = 0 \\ 2 α_{2} - 2 α_{3} = 0 \end{cases}$
Row-reduce the coefficient matrix
The coefficient matrix has columns indexed by $\alpha_1, \alpha_2, \alpha_3$ and rows given by the four entries. Reduction gives rank $r = 2$ , so there are $n - r = 3 - 2 = 1$ free variables.
$[\begin{matrix} 1 & 0 & 2 \\ 2 & 1 & 3 \\ - 1 & 3 & - 5 \\ 0 & 2 & - 2 \end{matrix}] ⟶ [\begin{matrix} 1 & 0 & 2 \\ 0 & 1 & - 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$
Conclude dependence and produce a witness
With one free variable the homogeneous system has non-trivial solutions, so the set is linearly dependent. The reduced form gives $\alpha_1 = -2\alpha_3$ and $\alpha_2 = \alpha_3$ . Choosing $\alpha_3 = -1$ gives $\alpha_1 = 2$ , $\alpha_2 = -1$ , and the resulting relation is verifiable entry by entry:
$2 B_{1} - B_{2} - B_{3} = [\begin{matrix} 2 & 4 \\ - 2 & 0 \end{matrix}] - [\begin{matrix} 0 & 1 \\ 3 & 2 \end{matrix}] - [\begin{matrix} 2 & 3 \\ - 5 & - 2 \end{matrix}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$
Identify the redundant block
The coefficient of $B_3$ is non-zero, so $B_3$ can be solved for: $B_3 = 2B_1 - B_2$ . The third stored block therefore adds nothing to the span of the first two and can be deleted without changing what the library can realise.
Test a repaired set
Replace $B_3$ by the identity matrix $I_2$ and repeat the procedure. Expanding $\alpha_1 B_1 + \alpha_2 B_2 + \alpha_3 I_2$ and equating entries gives a new homogeneous system:
${\begin{cases} α_{1} + α_{3} = 0 \\ 2 α_{1} + α_{2} = 0 \\ - α_{1} + 3 α_{2} = 0 \\ 2 α_{2} + α_{3} = 0 \end{cases}$
Solve the repaired system
The second equation gives $\alpha_2 = -2\alpha_1$ . Substituting into the third gives $-\alpha_1 - 6\alpha_1 = -7\alpha_1 = 0$ , so $\alpha_1 = 0$ , then $\alpha_2 = 0$ and $\alpha_3 = 0$ . The coefficient matrix has rank $3$ equal to the number of unknowns, so the trivial relation is the only relation of linear dependence and $\{B_1, B_2, I_2\}$ is linearly independent.

Result

The original library of three blocks has only two degrees of freedom: the third block is exactly $2B_1 - B_2$ and cannot extend the set of realisable couplings. Substituting the identity block restores full rank, so the repaired library of three blocks spans a three-dimensional subspace of the four-dimensional space $M_{22}$ and every realisable coupling has a unique set of gains. Uniqueness matters here beyond elegance: with the dependent set, gain values are not identifiable from measured behaviour, and any tuning or estimation procedure would wander along the one-parameter family of equivalent settings.

Applications & Industry Use

Control systems

Actuator redundancy and identifiability

A set of actuator influence vectors is dependent exactly when some actuator's effect can be reproduced by the others. The redundancy is valuable for fault tolerance but destroys identifiability: gains cannot be estimated uniquely from measured response, which must be resolved by a secondary allocation objective.

Structural engineering

Independent equilibrium equations

Nodal equilibrium equations for a truss are frequently dependent, and each dependency corresponds to a global equilibrium condition already implied by the others. Counting independent equations, rather than equations, is what gives the correct degree of static indeterminacy.

Signal processing

Redundant dictionaries

Overcomplete dictionaries are deliberately dependent, so representations are non-unique and a sparsity criterion selects among them. Knowing the dimension of the dependency — the number of free variables — quantifies how much choice the sparsity criterion has.

Chemical engineering

Independent reactions in a mechanism

A proposed reaction mechanism is dependent when one reaction is a combination of the others. Reducing to an independent subset gives the minimal set that reproduces every achievable composition change, which is the standard first step in mechanism reduction.

Experimental design

Valid contrast families

Contrast vectors in an analysis of variance must be independent for the corresponding effects to be separately estimable. A dependent contrast family produces an aliased design in which distinct effects cannot be distinguished from the data.

Statistics & machine learning

Perfect collinearity in a design matrix

Exactly dependent predictor columns make a regression unidentifiable: infinitely many coefficient vectors fit equally well. Detecting the dependency and removing one column restores a unique solution without changing the fitted values at all.

Design Considerations

Independence is a property of the set, not of any vector

No single vector is independent or dependent by itself in a multi-vector context. Statements such as "this vector is dependent" are shorthand for "this vector carries a non-zero coefficient in some relation on the set", and which vector plays that role is often a matter of choice. Report the relation, not a label on one member.

Pairwise checks do not establish independence

For a set of two vectors, dependence is equivalent to one being a scalar multiple of the other. For three or more, pairwise non-proportionality is entirely compatible with dependence — three coplanar vectors in $\mathbb{C}^3$ are a standard example. The full homogeneous system must be solved.

Choose the coordinate description before you start

The system's shape follows from the description: coefficients of a polynomial give one equation per power, entries of a matrix give one equation per entry. Fix the ordering once and use it consistently, since a silent reordering permutes the rows and can hide an arithmetic error rather than expose it.

Use the dimension bound as a free screen

In a space of dimension $d$ , any set of more than $d$ vectors is automatically dependent, because the coefficient matrix has more columns than rows. Checking $n$ against $d$ costs nothing and disposes of many candidate sets before any reduction begins.

In floating point, independence becomes conditioning

Exact independence is a yes/no property that rounding destroys. Numerically the meaningful question is how close the set is to being dependent, answered by the smallest singular value of the coordinate matrix or by its condition number. A set that is independent but nearly dependent behaves like a dependent one in any estimation task.

Decide whether redundancy is a defect or a feature

Dependence costs uniqueness of representation but can buy robustness, since the loss of one vector may leave the span intact. In dictionary design and in actuator layout, redundancy is deliberate. Establish which property the application values before removing dependent members.

Standards & Codes

Notation, interchange and numerical standards that govern how this material is written down, stored and computed in production systems.

Applicable standards, conventions and reference implementations
Reference	Title	Relevance to this topic
`ISO 80000-2`	Quantities and units — Part 2: Mathematics	Fixes the notation used for scalars, indexed vectors and the zero vector, and prescribes upright rendering for $\operatorname{rank}$ and $\dim$ .
`LAPACK / BLAS reference`	Linear Algebra PACKage reference implementation	Supplies the practical independence tests: `xGESVD` for a singular-value-based rank and `xGEQP3` for a pivoted factorisation that identifies which vectors are redundant.
`IEEE 754-2019`	IEEE Standard for Floating-Point Arithmetic	Defines the rounding that turns exact independence into a conditioning question, so any computed independence verdict is a statement about a tolerance rather than about the mathematics.
`ISO/IEC 40314`	Mathematical Markup Language (MathML) Version 3.0	Encodes the relations of linear dependence and matrix displays on this page as structured markup, keeping the coefficients machine-readable.
`W3C WCAG 2.1 AA`	Web Content Accessibility Guidelines	Requires the systems of equations and matrix arrays to be exposed semantically rather than as images, so their structure remains available at high zoom and to screen readers.

Material Selection

For a mathematical topic, "material" is the numeric representation: the scalar field, storage format and precision the computation is built from.

Representation and precision selection
Representation	Select when	Trade-off
Exact rational arithmetic	Certifying independence for symbolic or small integer sets, where the verdict must be unambiguous.	Gives a definite yes or no with no tolerance, but coefficient growth during reduction limits practical set sizes.
IEEE 754 binary64 with SVD	The default for numerical sets of measured or simulated vectors.	Robust and gives a graded answer via singular values, but costs more than an elimination and requires a documented cutoff.
Pivoted QR factorisation	Independence must be decided and the redundant members identified by name.	Cheaper than an SVD and returns a subset of the original vectors, but the rank decision is less reliable when singular values cluster.
Integer arithmetic modulo a prime	Rapid certification of independence for large integer coordinate matrices.	Exact and fast with bounded operand size, but an unlucky prime can falsely report dependence.
Dense coordinate matrix	Small to moderate sets in $P_n$ or $M_{mn}$ , where flattening to columns is natural.	Directly usable by optimised kernels, but discards structure such as symmetry or banding that the original objects had.
Sparse coordinate storage	Vectors with localised support, such as finite element shape functions or contrast codings.	Large savings in storage, but elimination fills in and an iterative or sparse QR method is needed to keep the advantage.

Manufacturing Notes

Implementation notes — how the result is actually produced by hand, by algorithm and by library, including cost and numerical behaviour.

Cost of the test

Testing $n$ vectors with $d$ coordinates each means reducing a $d \times n$ matrix, roughly $2dn^2 - \tfrac{2}{3}n^3$ operations for $d \ge n$ . A singular value decomposition of the same matrix costs a small constant multiple more and returns a graded measure of near-dependence, which is usually worth the difference on numerical data.

Doing it by hand

Expand the general relation completely before equating anything, and write the coefficient matrix with one column per vector and one row per coordinate. Because the system is homogeneous, the constants column is all zeros and can be omitted entirely — carrying it is a common source of transcription error and it can never become a pivot column.

Producing a witness relation

When free variables exist, set one to a convenient non-zero value such as $1$ or $-1$ and the rest to zero, then back out the dependent coefficients from the reduced rows. Clearing denominators gives an integer relation that is easy to verify by direct substitution, which is the form a witness should be published in.

Library behaviour

SymPy's Matrix.rank and Matrix.nullspace answer the question exactly, and an empty null space certifies independence. NumPy offers numpy.linalg.matrix_rank, which is SVD-based with a default tolerance derived from the matrix norm and dimensions; that default should be reviewed rather than accepted silently for ill-scaled data.

Scaling before testing

Independence is unaffected by scaling each vector, but the numerical rank decision is not. Normalising every coordinate vector to unit length before computing singular values removes an artificial spread caused by differing units and makes a single tolerance meaningful across the whole set.

Failure Modes & Common Mistakes

Failure modes, root causes and prevention
Failure mode / mistake	Impact	Root cause	Prevention & detection
Citing the trivial relation as evidence	high	Observing that all-zero coefficients give the zero vector and concluding something about the set.	State explicitly that the trivial relation always exists; only its uniqueness or non-uniqueness is informative.
Testing pairs instead of the whole set	high	Checking that no vector is a scalar multiple of another and concluding independence for three or more vectors.	Solve the full homogeneous system; pairwise proportionality is sufficient for dependence but never necessary.
Carrying an augmented column	medium	Treating the homogeneous system as a general system and reducing an augmented matrix whose last column is zero.	Reduce the coefficient matrix alone; a zero column stays zero and can never become a pivot column.
Concluding independence from a consistent system	high	Observing that the homogeneous system is consistent, which is automatic, rather than checking for free variables.	Compare rank with the number of vectors; independence requires rank equal to $n$ , not mere consistency.
Missing the zero vector in the set	medium	Running a full reduction on a set that contains $\vec{0}$ , which is dependent for a trivial reason.	Screen for the zero vector, duplicates and oversized sets before starting the computation.
Exact-zero rank test in floating point	high	Counting non-zero pivots on rounded coordinates, so a near-dependency is reported as independence or the reverse.	Use singular values with a documented tolerance and normalise the vectors first.
Mismatched coordinate ordering	medium	Equating coefficients in one order for some vectors and another order for others, producing a permuted and incorrect system.	Fix a single coordinate ordering and record it beside the coefficient matrix.
Assuming independence survives a transformation	low	Applying a linear map to an independent set and assuming the images remain independent; only injective maps guarantee this.	Check that the map has trivial kernel before transferring an independence claim through it.

FAQs

Why is the definition phrased in terms of the zero vector rather than one vector being a combination of the others?

The two are equivalent, but the zero-vector formulation is symmetric in all the members and translates directly into a homogeneous system. The alternative formulation would require checking each vector in turn against the others, which is $n$ separate consistency questions rather than one rank computation.

Is a set containing the zero vector ever independent?

No. Take coefficient one on the zero vector and zero on everything else. The combination equals the zero vector, and not all coefficients are zero, so this is a non-trivial relation of linear dependence. Any set containing $\vec{0}$ is therefore dependent regardless of what else it contains.

How many vectors can an independent set contain?

At most the dimension of the space. In $P_4$ , which has dimension five, no six polynomials can be independent; in $M_{22}$ , which has dimension four, no five matrices can be. The reason is that the coefficient matrix would then have more columns than rows and could not attain full column rank.

Why is proving independence harder than proving dependence?

Dependence is an existence claim, settled by producing one witness that anyone can check. Independence is a universal claim — no non-trivial relation exists — and universal claims cannot be settled by examples. The homogeneous system's general solution is what discharges the universal quantifier.

Does linear independence in a polynomial space mean the polynomials have no common roots?

No, the two notions are unrelated. The polynomials $x - 2$ , $(x-2)^2$ and $(x-2)^3$ share the root $x = 2$ and are linearly independent. Conversely two polynomials with no common root can be dependent if one is a scalar multiple of the other, which is impossible only when the multiple is non-trivial.

What does a free variable in the homogeneous system actually tell me?

Each free variable is one independent direction of redundancy in the set. With $n$ vectors and rank $r$ , there are $n - r$ free variables, and $n - r$ vectors can be removed — chosen appropriately — without changing the span. The free variables therefore quantify exactly how much of the set is surplus.

How should I decide independence for numerical data?

Assemble the coordinate matrix, normalise the columns, compute the singular values, and compare the smallest with the largest. Exact independence is not a stable property under rounding, so report the condition number or the smallest singular value alongside any verdict, and state the tolerance used.

References

Beezer, R. A. A First Course in Linear Algebra, Version 0.70. University of Puget Sound, 2006. Section B, Subsection LI, Definitions RLD and LI. Licensed under the GNU Free Documentation License v1.2.
Axler, S. Linear Algebra Done Right, 3rd edition. Springer, 2015.
Golub, G. H. and Van Loan, C. F. Matrix Computations, 4th edition. Johns Hopkins University Press, 2013.
ISO 80000-2:2019, Quantities and units — Part 2: Mathematics. International Organization for Standardization.
IEEE 754-2019, IEEE Standard for Floating-Point Arithmetic. Institute of Electrical and Electronics Engineers.

AI Suggested Questions

Give me three vectors in $\mathbb{C}^3$ that are pairwise non-proportional yet linearly dependent, and show the relation.
Show that $\{x - 2, (x-2)^2, (x-2)^3\}$ is linearly independent in $P_3$ despite sharing a common root.
For a set of five matrices in $M_{22}$ , explain why dependence is guaranteed before any computation is done.
How do I use a pivoted QR factorisation to identify which vectors in a dependent set are redundant?
What is the relationship between the smallest singular value of a coordinate matrix and near-linear-dependence?
Explain why linear independence is preserved by an injective linear transformation but not by an arbitrary one.

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