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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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Nielsen & Chuang, Problem 2.2 — Properties of the Schmidt number

I started reading the Nielsen & Chuang's Quantum Computation and Quantum Information. I got stuck by the last question of Problem 2.2. I got the other problems, but I can't see this one. I guess it's not really difficult, but as I am new in this…
fay
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linear algebra basic proof

F is a linear functional in $V'$ a linear vector space which operates on $\phi\in V$. Show that there is a one-to one correspondence between F and $f\in V $ such that $F(\phi)=(f,\phi)$ where $V$ is also a linear vector space. $(,)$ represents…
Please Delete Account
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Can a subspace be written as the direct sum of its intersections with subspaces generated by partitions of a basis?

Say $V$ is a vector space with some basis $\mathcal{B}=\{b_i\}_{i\in I}$, and let $\{B_1,\dots,B_n\}$ be a partition of $\mathcal{B}$. In general, for $S$ a subspace it is not true that $$ S=\bigoplus_{i=1}^n (S\cap\langle B_i\rangle). $$ For…
Noomi Holloway
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Are there infinitely many $A\in \mathbb{C}^{2 \times 2}$ satisfying $A^3 = A$?

Let $A$ be a $2 × 2$-matrix with complex entries. The number of $2 × 2$-matrices $A$ with complex entries satisfying the equation $A^3 = A$ is infinite. Is the above statement true? I know that $0$ and I are two solutions. But are there any more…
TUMO
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Linear Algebra - basis question

I am revising for a Linear Algebra exam by going through some previous quiz questions, that I have True/False answers to, but not the reasoning or counterexamples. I am stuck on the following: If $v_1,v_2,v_3,v_4$ is a basis for $V$, and $U$ is a…
ghufran syed
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$n+1$ points in $\mathbb{R}^n$ with pairwise rational distances are linearly dependent

Let $v_0$ be the zero vector in $\mathbb{R}^n$ and let $v_1, v_2, . . . , v_{n+1}$ be vectors in $\mathbb{R}^n$ such that the Euclidean norm $|v_i − v_j|$ is rational for every $0 ≤ i, j ≤ n + 1$. Prove that $v_1, . . . , v_{n+1}$ are linearly…
math_lover
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A $3 \times 3$ matrix with one eigenvalue and one eigenvector?

Suppose we have to construct a $3 \times 3$ matrix with only one eigenvalue which has only one linearly independent eigenvector, what should be our approach? I was asked this in an interview, so first thing that came on my mind was to look for a…
blabla
  • 1,104
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Linearly independent subset of polynomials

Let $S$ be a set of non-zero polynomials over a field $F$. If no two elements of $S$ have the same degree, show that $S$ is an independent subset of $F[x]$. I tried to prove the statement by induction on the number of elements of any finite subset…
luimichael
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Why is it that when a determinant = 0 then the homogeneous equations represented by the matrix has a non trivial solution?

As I was following a lecture the instructor seemed to assume this and when on solve for the equations where the right side was equal to 0 and proceed with the problem but I know if a determinant is non zero than an inverse matrix exists and visa…
Sedumjoy
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Rank nullity theorem -bijection

Let $T:\Bbb R^n\to \Bbb R^n$ be a linear transformation. Which of the following statement implies that $T$ is bijective? a) $\operatorname{Null}(T)=n$ b) $\operatorname{Rank}(T)=\operatorname{Null}(T)=n$ c)…
user450820
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Basis for Kernel and Image of the following T

I am working on this practice problem, and I was wondering if I could get some help. I have a $T$:$\mathbb{R^{2x2}}\to \mathbf{P_{2}}$, that is, from 2x2 matrices to polynomials of degree at most 2. The transformation is given as following:…
LinAlgStudent
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Basic trace inequality

Suppose $A$ and $B$ are self-adjoint matrices. Why is it true that $$Tr(A^2) \le Tr(Ae^{-tB}Ae^{tB})$$ for $t\in\mathbb R$, where $e^x$ denotes the matrix exponential?
Potato
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Characteristic Polynomial Independent From the Choice of Basis

I want to prove that definition of characteristic polynomial of a linear operator on a finite-dimensional vector space $V$ is independent of the choice of basis for $V$. Proof>> Choose two different basis for V, $\beta, \beta '$. Let $Q =…
Beverlie
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A question about commutativity of linear operators

Let $V$ be a finite dimensional vector space over $F$ and $T$ be a linear operator defined on $V$. Then if $p(t)$ and $q(t)$ are polynomials defined over $F$, then the the operators $P(T)$ and $Q(T)$ commute. Is the converse true? i.e $AB=BA…
James
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Why define three elementary row operations? When one of them can be performed by the others?

I'm learning about linear algebra and in the course we've defined three "elementary row operations" $$(1) \text{ Switching any two rows}$$ $$(2) \text{ Non-Zero scaling of any row}$$ $$(3) \text{ Adding a multiple of one row to a different…
nathan56
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