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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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if matrix such $AA^T=A^2$ then $A$ is symmetric?

let matrix $A_{n\times n}$ is real matrix,such $AA^T=A^2$, The transpose of matrix $A$ is written $A^T$, show that : the matrix $A$ is Symmetric matrices maybe this problem have more methos,because it is know that if matrix $A$ is symmetric,then we…
math110
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What does "extend linearly" mean in linear algebra?

The following proposition is from one of Gowers's articles: Let $X$ be a vector space, and $x\in X$, $x\neq {\bf 0}$. Then there exists a linear map $g:X\to {\mathbb R}$ such that $g(x)\neq 0$. The existence of this map can be proved as follows.…
user9464
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Express $\mathrm{Tr}(X)$ in terms of $A$, given that $X=A^TX(I+X)^{-1}A$

Given real non-singular $n\times n$ matrix $A$, with all eigenvalues larger that $1$. Express $\mathrm{Tr}(X)$ in terms of $A$, given that $X=A^TX(I+X)^{-1}A$. $\quad$($X$ is sym. pos. def.) It is allowed to assume that $A$ is in any special…
Lee
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Diagonalizable upper triangular matrices

It is true that if an upper triangular matrix $A$ with complex entries has distinct elements on the diagonal, then $A$ is diagonalizable. However, I don't think the converse is true. Is there a complete characterization of all diagonalizable upper…
user62727
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Is there something deep in the fact that an endomorphism of a finite dimensional complex vector space has an eigenvector?

In my course of linear algebra I studied that if $V$ is a finite dimensional vector space on the complex field, then every endomorphism of $V$ has an eigenvector. The proof is simple: taken a polynomial $f$ that is null on the endomorphism $A$ (it…
user365
13
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3 answers

Largest eigenvalue of a real symmetric matrix

If $\lambda$ is the largest eigenvalue of a real symmetric $n \times n$ matrix $H$, how can I show that: $$\forall v \in \mathbb{R^n}, ||v||=1 \implies v^tHv\leq \lambda$$ Thank you.
user8837
13
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invertible if and only if bijective

I need to show that a linear transformation $T \in L(W,W)$ is invertible if and only if $T$ is bijective. I'm not sure how to go about this, can someone give me an idea where to start? I know is injective if $T_u = T_w$ aka null $T = {0}$,…
rw173
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Why does the subspace need to go through the origin?

I understand that the main difference between a subspace and a hyperplane is that the subspace must go through the origin. Why does need to happen? In other words, why does a subspace always have to go through the origin? What restricts it from…
usr
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4 answers

Find the span of a set of vectors

In general, what is the most straightforward way to find the span of a set of vectors? I'm trying to find the span of these three vectors: $$\{[1, 3, 3], [0, 0, 1], [1, 3, 1]\}$$
Anderson Green
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Is $\mathbb R^2$ a field?

I'm new to this very interesting world of mathematics, and I'm trying to learn some linear algebra from Khan academy. In the world of vector spaces and fields, I keep coming across the definition of $\mathbb R^2$ as a vector space ontop of the field…
vondip
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Example of product space isomorphic to sum of subspaces

Here is the problem statement (from chapter $3$ of Axler's Linear Algebra Done Right). Give an example of a vector space $V$ and subspaces $U_1,U_2$ of $V$ such that $U_1 \times U_2$ is isomorphic to $U_1 + U_2$, but $U_1 + U_2$ is not a direct…
Daniel Xiang
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Is $1 = [1] = [[1]] = [[[ 1 ]]], \ldots$?

Is the following true? Where [a] is a 1x1 matrix containing the object…
Dmytro
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A "geometric'' infinite sum of matrices

The sum $$ I + A + A^2 + A^3 + \cdots $$ equals $$(I-A)^{-1}$$ under the assumption $\rho(A)<1$, which is necessary to make the sum converge. My question: what does the sum $$ I + A^T A + (A^2)^T A^2 + (A^3)^T A^3 + \cdots$$ equal under the same…
calc student
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What does $\mathbb{R}^n \to \mathbb{R}^m$ mean? And what is $\mathbb{R}^n$?

What the does $\mathbb{R^n}$ mean? For example if something says that it is a transformation $T:\mathbb{R}^2 \rightarrow \mathbb{R}^3$. Does that mean that $\mathbb{R}^2 = 2 \times 2$ matrix? and that $\mathbb{R}^3 = 3 \times 3 $ matrix?
Yusha
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proof on trace trick

I wanted to find the maximum likelihood estimator for $\mathbf{\Sigma}$ in the multivariate gaussian. I was anticipating the solution would be a bit involved and messy, if not 'brute-forced', but I was surprised to find an elegant and clever…
cgo
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