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Questions tagged [linear-algebra]

For questions on linear algebra, including vector spaces, linear transformations, systems of linear equations, spanning sets, bases, dimensions and vector subspaces.

Linear algebra is concerned with vector spaces and linear transformations between them:

$$(x_1, \dots, x_n)\to a_1x_n+\dots+a_nx_n$$

Concepts include systems of linear equations, bases, dimensions, subspaces, matrices, determinants, kernels, null spaces, column spaces, traces, eigenvalues and eigenvectors, diagonalization and Jordan normal forms.

This is a general tag, most of the subjects included have secondary tags, e.g.

127034 questions
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Closest point on a line to another point

The problem asks to find the nearest to $P(2,2)$, on the line: $$-x -2y +3 =0$$ This is what I've tried, the normal vector, $$n = (-1,-2)$$ found an arbitrary point on the given line by setting $y = 0$, which results in $$P_1 = (3,0)$$ calculated…
JLL
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Why does linear dependence require a *finite* linear combination to vanish?

By definition, $S$ is linearly independent if for all $n>0,c_i\in F,s_i\in S$, we have that $c_1s_1 + \ldots + c_n s_n \neq 0$ whenever $\{c_i\}\neq \{0\}$. Why do we restrict our attention to finite linear combinations? We could imagine a different…
tba
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Intuition for cross product of vector with itself and vector with zero vector

I'm having trouble intuiting the following two vector identities for any vector $\mathbf{v}$. I'm only asking about intuition here, and not about their proofs (which follow from definition of cross product): $\color{green}{\mathbf{v}} \times…
user53259
16
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2 answers

Does zero vector have zero dimension?

I know this sounds like a stupid question, but I just want to organize and clear what I studied. For an $n\times n$ matrix $A$, it has independent columns when nullspace only has zero vector. And independent columns mean $A$ has rank $n$,…
email
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Span of a subset of a vector space is the smallest subspace containing that set

To Prove: If $S=[{v_1,v_2,...,v_k}]$ is a subset of vector space $V$. Then $span(S)$ is the smallest subspace of $V$ containing set $S$. I know that $L[S]$ is a subspace of $V$. But in most arguments for proving the $L[S]$ is the smallest subspace…
Jyotishraj Thoudam
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16
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3 answers

Can an idempotent matrix be complex?

A matrix $A$ is called idempotent if $A^2 = A$. I am just wondering if such matrix can be complex. Anyone can help give an example or proof that it has to be real? Thanks!
Tuyet
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Prove that $0<\det(A) \le 1$

$A=(a_{ij})$ is a $n\times n$ symmetric real matrix such that: $a_{ii}=1$ and $\sum_{j=1}^{n}|a_{ij}|<2$ for all $i \in \{1,2,3,...,n\}$. Prove that $0< \det(A) \le 1$. My approach: That is a question that I have tried before and I am trying again…
Arnaldo
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Simultaneously diagonalisable iff A,B commute

Yes, this is a repeat, however I have not seen anyone explain it fully (or such that I can comprehend it, and believe me, I have searched thoroughly for answers). If the (linear) endomorphisms $A,B: V \to V$ are diagonalisable, show that they are…
user305860
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3 answers

Is there a name for a matrix where the column space equals the row space? Does it have any interesting properties?

I sat down to write a linear algebra take-home exam problem where I would give a $4\times4$ matrix $A$ and ask for bases for these six spaces: $\mathrm{Col}\,A$, $\mathrm{Row}\,A$, $\mathrm{Nul}\,A$, $\mathrm{Col}\,A^t$, $\mathrm{Row}\,A^t$,…
2'5 9'2
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a 2 distance set has an upper bound for number of its elements

Can anyone help me with this problem? We call a set $A\subseteq \mathbb R^n$ a $2$-distance set if for each $v_i,v_j$ in $A$, $i\neq j$, $|v_i-v_j|=r$ or $s$. Find an upper bound for the number of the elements of $A$. Our Teacher proved that $|A|\le…
Goodarz Mehr
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Linear functional equation

During my mathematical musings I encountered the following functional equation : denote by $L$ the set of all functions ${\mathbb Z}^2 \to {\mathbb C}$ satisfying $$ \begin{array}{cl} &f(x+a,y+b)+f(x+b,y+c)+f(x+c,y+a) \\ =&…
Ewan Delanoy
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2 answers

Is every endomorphism of a vector space a linear combination of idempotents?

Is every endomorphism of a $K$-vector space a $K$-linear combination of idempotents? This question was asked by Jonas Meyer in a comment to this question. To make sure I earn no points thanks to a question raised by somebody else, I shall put a…
Pierre-Yves Gaillard
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Express the following invertible matrix A as a product of elementary matrices

I've been at this for a while... I tried to the inverse method but it keeps on saying I'm getting it wrong... Can anyone show me a step-by-step solution? The matrix I have is a $3\times 3$ square one(sorry for formatting): $$ \begin{pmatrix} 6 & 6…
karinamel
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6 answers

Proof of Gaussian elimination/Why does it work

I have just had a class on linear algebra and the professor explained how to solve matrixes. While he could explain how to solve them by using Gaussian's elimination, he failed to explain how does that work. Why does matrix before doing any…
khajvah
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15
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2 answers

Does every vector space contain a zero vector?

If this is true, then every vector space must always have at least one subspace, the one consisting of only the zero vector, correct? Thanks!
Johnathon Svenkat
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