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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 a $3 \times 3$ matrix $A$ is similar to $A^2$, find all possible Jordan Canonical Forms of $A$

I want to find all possible Jordan Canonical Forms (JCF) of $3\times 3$ matrix $A$, if $A$ is similar to $A^2$. I have only $4$, and I think there is something wrong with my solution. Let $J$ be JCF of $A$, then $J^2$ is JCF of $A^2$, since $A$…
Jennie
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Can the transpose of a matrix be expressed in row/column operations?

Suppose that $A$ is a matrix, can we get its transpose, $A^T$, by performing row and/or column operations to $A$?
Not an ID
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A question about the proof of "a symmetric matrix has real eigenvalues".

This question is in reference to this proof of the Spectral Theorem. Let $A$ be a symmetric matrix. Take $u$ to be an orthonormal vector (such that $u^Tu=1$). Then if $u$ is an eigenvector of $A$, then $Au=\lambda u$. This implies that…
algebraically_speaking
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Is $\mathbb{R}$ a subspace of $\mathbb{R}^2$?

I think I've been confusing myself about the language of subspaces and so on. This is a rather basic question, so please bare with me. I'm wondering why we do not (or perhaps "we" do, and I just don't know about it) say that $\mathbb{ R } $ is a…
aherring
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Vectors spanning a plane

I am having a problem with this question, I just can't seem to get it. Consider the plane $\mathbb{R}^3$ defined by the equation $x+2y-z=0$ Find any two vectors $\mathbf{v},\mathbf{w}$ such that their span may be identified with this plane.
enawang
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How to find adjoint of linear operator T on inner product space V

Let $V$ be an inner product space and $T$ a linear operator with $T(\alpha) = (\alpha,\beta)\gamma$ for fixed elements $\beta,\gamma \in V$. I now that $T$ is linear operator. How we can show that adjoint of $T$ ($T^*$) exist and what is it?
SKMohammadi
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Verifying the examples of Dual Space

I am new to the concept of Dual space. Can someone please explain which of these are dual spaces and why? Also if $y$ is a polynomial on $x$, then what does $x(0)$ means? • For $x \in P$, $y(x) = x(0)$ • Any vector space V, $y(x) = 0$; for every $x…
Manish
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Pairwise commuting nilpotent matrices: alternative solution needed

I have a problem: Let $A_1,A_2,...,A_n$ be $n\times n$ nilpotent matrices which are commute in each pair ($A_iA_j=A_jA_i$). Prove that: $$A_1A_2...A_n=0$$ I have got a solution by proving that $Im(A_n)$ is an invariant supspace under…
anonymous67
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Dimension of the sum of three subspaces

We know that $$\dim(U_1 + U_2) = \dim U_1 + \dim U_2 - \dim(U_1 \cap U_2)$$ if $U_1$ and $U_2$ are finite dimensional subspaces. For three finite dimensional subspaces prove or give a counterexample for the following: $$ \begin{align} \dim(U_1 + U_2…
Soaps
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Linear operator T that commutes with every projection operator, infinite dimensional case

This is related to a question of Hoffman & Kunze, Linear Algebra (Section 6.7, #8, p. 219). The question in the text asks: Let $T$ be a linear operator on $V$ which commutes with every projection operator on $V$. What can you say about $T$? If…
kalam
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Distance of a vector from a subspace - Linear Algebra

I want to calculate the distance of the vector $x=(1,1,1,1)$ to the subspace $\{(1,0,2,0) , (0,1,0,2)\}$ I have solved this in 2 ways that I know of but the thing is, the results are different. For instance when I use $||x-Pr(x)||$ I get…
Kyle
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Determine whether A is invertible, and if so, find the inverse. (3x3)

In Exercises 37-38, determine whether $A$ is invertible, and if so, find the inverse. [Hint: Solve $AX = I$ for $X$ by equating corresponding entries on the two sides. 37. $A = \begin{bmatrix} 1&0&1 \\ 1&1&0 \\ 0&1&1 \end{bmatrix}$ How the heck am…
J L
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Singular Values/l2-norm of Pseudo-inverse

I am trying to prove, given a matrix $A=\lbrack\frac{A_1}{A_2}\rbrack\in C^{m\times n}$, with $A_1\in C^{n\times n}$ non-singular, that: $||A^+||_2\leq||A_1^{-1}||_2$ ($||\cdot||_2$ is the induced $\ell_2$ norm, $(\cdot)^+$ is the Moore-Penrose…
John436
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Proving $(A+B)^2=A^2+2AB+B^2$ wrong using examples

Give a counter example to show the following statements are false? $$(A+B)^2=A^2+2AB+B^2.$$ Would it be possible to use the $2\times 2$ matrices to show the statement the above?
shaad
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Correspondence between two matrices

Suppose $B$ is a positive definite matrix with determinant $1 $ and $$ A = \frac{1}{2} \int_0^\infty \frac{(B+sI)^{-1}}{\sqrt{\mbox{det}(B+sI)}} ds $$ Then, how does one prove that this provides a one to one onto correspondence between positive…
Shibi Vasudevan
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