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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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Prove properties of $A^2 = -I$

Given an $n\times n$ matrix A with real entries such that $A^2=-I$, prove (a) that $n$ is even and (b) that $A$ has no real eigenvalues. How do you do this? I have no idea where to start.
john
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Proving Distributivity of Matrix Multiplication

If $A,B,C$ are matrices I am thinking how to show that $$ A(B + C) = AB + AC$$ Is possible to show without sums like $\sum_i a_i, ..., \sum_j b_j$? It seems if I do the proof with many indexes then is tedious and I don't learn much from it.
blue
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Proving if $v, T(v)\, ..., T^{k}(v)$ are linearly dependent for every $v$, then $I, T, ..., T^{k}$ are linearly dependent.

Suppose $V$ is a finite-dimensional vector space. Take a linear operator $T \in L(V)$. Now suppose that we know for every $v \in V$, the set of vectors $\{v, T(v)\, ..., T^{k}(v)\}$ is linearly dependent. I want to show that this would imply that…
Ransplito
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How to distinguish between vertical and horizontal stretch/shrink when ambiguous?

Please bear with me. I am trying to help my daughter with her Algebra 1 homework. We are asked to describe the transformation of function f to function g as follows: $$f(x) = x$$ $$g(x) = 2x+3$$ The provided answer states that $g(x)=2x+3$ can be…
Mark Featherstone
  • 183
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If $A$ is an $n \times n$ matrix and $ A^2 = 0$, then $\text{rank}(A)\le n/2$.

Was revising for a Linear Algebra when I came across this question. "Given a Matrix $A$ over $\mathbb{R}^{n \times n}$, and $A^2 = 0$, show that rank($A$) $\leq n/2$" My attempt: I was thinking of the theorem from Cayley-Hamilton but the matrix…
Jon Gan
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If a vector space and one subspace are both infinite-dimensional. Are they equal?

From linear algebra I know this theorem: If V is a finite-dimensional vector space and W is a subspace of V, then $$ i)~ dim~ W \leq dim ~V\\ ii) ~dim~ W = dim~ V \longrightarrow W=V $$ As a lemma: If $dim ~W = \infty$ $\longrightarrow$ $dim~V =…
Spherk
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Outer product of a vector with itself

Is there a special name for an outer product of a vector with itself? Is it a special case of a Gramian? I've seen them a thousand times, but I have no idea if such product has a name. Update: The case of outer product I'm talking about is…
Phonon
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What are the relations between eigenvectors of $A$ and its adjoint $A^*$?

Everywhere I can read that a matrix and its adjoint have pretty much the same eigenvalues ( only complex conjugation is the difference between them). Now I was wondering whether such a relation also exists between the eigenvectors of both matrices.…
user66906
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Geometric interpretation of $\mathbb R^n$

Motivated by this question Basis of a basis I've been thinking when we say that a vector can have a geometrical interpretation, are we talking about the vectors themselves or the vectors coordinates in the usual basis $\{(1,0),(0,1)\}$ in for…
user42912
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Does having real eigenvalues have a geometric meaning?

Is it possible to characterize the set of real matrices which have real eigenvalues geometrically? That is, is it possible to say that a linear map $T$ has real eigenvalues if and only if it satisfies some property ${\cal P}$ which has a geometric…
aro
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If $A$ is orthogonal, for any vector $x$ such that $Ax = b$, $\Vert x \Vert = \Vert b \Vert$

Is this statment true: For any vector $x$ such that $Ax = b$, $\Vert x \Vert = \Vert b \Vert$, if $A$ is orthogonal. I was working on a proof for my linear algebra class, when I noticed that the entire proof could be reduced to simple algebraic work…
Alexander Ventura
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Can infinite-dimensional vector spaces be decomposed into direct sum of its subspaces?

I'm reading Axler "Linear agebra done right" and in Chapter 1 he discusses subspaces and direct sum. My question is, are there subspaces of the infinite-dimensional vector spaces, e.g. a functional Banach space with sup norm $V$ that directly sum to…
Bayes
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Which vectors have unique representation when it isn't a direct sum?

I know that $V = U \oplus W$ means that every $v \in V$ can be written uniquely as $v = u + w$ for some $u \in U, w \in W$. However, what happens if $V = U + W$ is not direct? Then this means that some vector $v$ does not have a unique…
twosigma
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How fast can you determine if vectors are linearly independent?

Let us suppose you have $m$ real-valued vectors of length $n$ where $n \geq m$. How fast can you determine if they are linearly independent? In the case where $m = n$ one way to determine independence would be to compute the determinant of the…
Martin Drozdik
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Contradiction in the rules regarding determinants and row operations?

In my textbook it says that if you multiply a row in a matrix $A$ by a nonzero constant $c$ to obtain $B$, then $\det{B}=c\det{A}$. Later on it says that if you obtain $B = cA$ by adding $c$ times the $k^{\text{th}}$ row of $A$ to the…
ithisa
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