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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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Linear Algebra, cube & dimensions > 3

I have found interesting problem in Gilbert's Strang book, ,,Introduction to Linear Algebra'' (3rd edition): How many corners does a cube have in 4 dimensions? How many faces? How many edges? A typical corner is $(0,0,1,0)$ I have found the answer…
exTyn
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Proof relating inverse to determinant

I'm reading a paper regarding the consistency of a statistical estimator, and the author claimed that the following is an identity: $$ \mathbf{x}^\top (\Sigma + \mathbf{x}\mathbf{x}^\top)^{-1}\mathbf{x} = 1- \frac{\det (\Sigma)}{\det…
firdaus
  • 703
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2 answers

Show trace is zero

Problem: We are given $n\times n$ square matrices $A$ and $B$ with $AB+BA=0$ and $A^2+B^2=I$. Show $tr(A)=tr(B)=0$. Thoughts: We have $tr(BA)=tr(AB)=-tr(BA)=0$. We also have the factorizations $(A+B)^2=I$ and $(A-B)^2=I$ by combining the two…
Potato
  • 40,171
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3 answers

Are the functions $\sin^n(x)$ linearly independent?

The following problem is from Golan's linear algebra book. I have posted a proposed solution in the answers. Problem: For $n\in \mathbb{N}$, consider the function $f_n(x)=\sin^n(x)$ as an element of the vector space $\mathbb{R}^\mathbb{R}$ over…
Potato
  • 40,171
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2 answers

Why is a matrix $A\in \operatorname{SL}(2,\mathbb{R})$ with $|\operatorname{tr}(A)|<2$ conjugate to a matrix of the following form?

The trace $\operatorname{tr}(A)$ of a matrix $A$ is the sum of its diagonal entries. Apparently if $A\in \operatorname{SL}(2,\mathbb{R})$ and $|\operatorname{tr}(A)|<2$, then $A$ is conjugate in $\operatorname{SL}(2,\mathbb{R})$ to a matrix of the…
Tara B
  • 5,341
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Show that $B$ is invertible if $B=A^2-2A+2I$ and $A^3=2I$

If $A$ is $40\times 40$ matrix such that $A^3=2I$ show that $B$ is invertible where $B=A^2-2A+2I$. I tried to evaluate $B(A-I)$ , $B(A+I)$ , $B(A-2I)$ ... but I couldn't find anything.
Mücahit Meral
  • 351
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1 answer

What exactly is antieigenvalue analysis?

I found a book in the library about antieigenvalue analysis and it is possibly the most unreadable piece of literature I have ever made an effort to understand. Unfortunately, every other resource I try inevitably takes you back to the same…
JessicaK
  • 7,655
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Show whether matrix is positive semidefinite or not

Background and motivation: When creating a Mercer Kernel Function we need to show that the Gram matrix defined by the function is positive semidefinite. Let $A_1, \ldots, A_n$ be subsets of $\{0, 1, \ldots, D\}$. Let $S(X)$ be the smallest $k$…
soerend
  • 131
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3 answers

Eigenvalues of product matrix

I have two matrices, both positive definite, real symmetric and one is diagonal. What can I say about lower and upper bound of the eigenvalues of the product matrix in terms the of lower and upper bounds on eigenvalues of those two matrices.
user5644
13
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2 answers

Adjoint Operator and Inverse

I am solving the following question and I am not really sure about the way I approach Question 1: Assume that $T:U\rightarrow U$ is invertible map. Prove that $(T^*)^{-1}=(T^{-1})^*$ Here is my answer: Notice that $\langle Tv,u\rangle = \langle v,…
needhelp
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Why isn't the quotient space $V/V = \{ V \}$?

If $W \subset V$, then one defines the quotient space, $$V/W = \{ v + W : v \in V \}$$ So why isn't this right? $$V/V = \{v + V : v \in V \} = \{V \}$$? I read that $V/V = \{ 0 \}$? Why can't the whole set $V/V$ be partition, by $V$ itself?
Lemon
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5 answers

$T$ be linear operator on $V$ by $T(B)=AB-BA$. Prove that if A is a nilpotent matrix, then $T$ is a nilpotent operator.

Let $V$ be a vector space of $n\times n$ matrices over a field F, and let $A$ be a fixed $n\times n$ matrix. $T$ be linear operator on $V$ by $T(B)=AB-BA$. Prove that if A is a nilpotent matrix, then $T$ is a nilpotent operator. I have done in a…
Ri-Li
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1 answer

What do adjoints have to do with this problem?

Question: Let $V\ $ be the vector space of the polynomials over $\mathbf{R}$ of degree less than or equal to 3, with the inner product $$ (f|g) = \int_0^1 f(t)g(t) dt. $$ If $t$ is a real number, find the polynomial $g_t$ in $V$ such that $(f|g_t) =…
zrbecker
  • 4,048
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2 answers

The meaning of Inverse Matrix

I am studying Linear Algebra, I have 3 questions in my mind What does an inverse matrix mean. I am trying to have a meaning of it, but I don't really understand. When a matrix does not have an inverse matrix, what does it mean? Hope to hear your…
user122358
  • 2,712
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3 answers

Definition of Trace of Linear Operator

The trace of a linear operator $f$ can be defined as the trace of the matrix $A$ representing $f$ with respect to some basis $B$. However the trace does not depend on the basis chosen. This suggests to me that there is some definition of the trace…
user137731
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