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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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Can vector spaces over different fields be isomorphic?

Two vector spaces are said to be isomorphic iff there's an invertible linear map between them. It can be shown that isomorphic vectors spaces would have to have the same finite dimension or both be infinite dimensional. But what if they are over…
Nishant
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Singular value decomposition of product of matrices

Given SVD(A) and SVD(B) and B is a diagonal matrix, is there a way or method to construct SVD(AB) ?
user16409
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4 answers

Null space for $AA^{T}$ is the same as Null space for $A^{T}$

$A$ is an $n\times m$ matrix and $AA^{T}$ is a symmetric real matrix. Also, we have: $\operatorname{rank}(AA^{T})=r\stackrel{?}{=}\operatorname{rank}(A)$. Let $Q= \begin{Bmatrix} q_1,...,q_{n-r} \end{Bmatrix}$ be a basis for the Null space of…
Austin
  • 303
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$A\in M_3(\mathbb R)$ orthogonal matrix with $\det (A)=1$. Prove that $(\mathrm{tr} A)^2- \mathrm{tr}(A^2) = 2 \mathrm{tr} (A)$

$A\in M_3(\mathbb R)$ orthogonal matrix with $\det (A)=1$. I need to prove that $(\mathrm{tr} A)^2-\mathrm{tr}(A^2) = 2 \mathrm{tr} (A)$ ; $\mathrm{tr}$=trace. I know that if $A$ is orthogonal than $A^tA=I$ and that $A$ is diagonalizable and similar…
Jozef
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Equation of plane containing two vectors

I am struggling with the interpretation of this question: Vectors: $u = \left(1,\ 0,\ \sqrt3 \right)$ and $v = (1,\ \sqrt3,\ 0)$ in standard position. Find an equation of the plane containing $u$ and $v$. Am I correct in interpreting this question…
Will777
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Linear Dependence Lemma

This is out of my textbook, Axler's "Linear Algebra Done Right" which I am self-studying from. (I organized my thoughts in which I would like some sort of response with Roman Numerals). Linear Dependence Lemma: If $(v_{1},\ldots,v_{m})$ is linearly…
St Vincent
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Prove that $\det(A)\neq 0$.

Let $A$ be a $n \times n$ matrix, $n$ even, with even diagonal elements and all other elements odd integers. Prove that $\det(A)\neq 0$. Can anyone give me a hint? Thank you.
user62138
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Dual spaces isomorphic, implies vector spaces itself are isomorphic?

When I have two vector spaces $W, V$ over $k$ a field. And I know that the algebraic dual spaces of $V$ and $W$ are isomorphic. Can I conclude, (in the infinite dimensional case) that $V$ and $W$ are isomorphic? I am saying algebraic dual, because i…
Fabio Neugebauer
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What exactly does linear dependence and linear independence imply?

I have a very hard time remembering which is which between linear independence and linear dependence... that is, if I am asked to specify whether a set of vectors are linearly dependent or independent, I'd be able to find out if $\vec{x}=\vec{0}$ is…
Mirrana
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5 answers

How to prove that the set $\{\sin(x),\sin(2x),...,\sin(mx)\}$ is linearly independent?

Could you help me to show that the functions $\sin(x),\sin(2x),...,\sin(mx)\in V$ are linearly independent, where $V$ is the space of real functions? Thanks.
Pedro
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Rank of a matrix

If a $3 \times 3$ matrix has determinant zero, then is it possible that its rank could be $3$? I think it only could be $2$ or less. I am right or wrong? Please explain.
Max
  • 111
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Finding a invariant subspaces for a specific matrix

Good morning, How does one find the subspaces that are invariant under $A$ for $$A = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 &2 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 3 \end{pmatrix}\ \in M_{4} (\mathbb{R}).$$ Thank you.
user6163
11
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4 answers

Given two subspaces $U,W$ of vector space $V$, how to show that $\dim(U)+\dim(W)=\dim(U+W)+\dim(U\cap W)$

Let $U,W$ be subspaces of a vector space $V$. Show that $$\dim(U)+\dim(W)=\dim(U+W)+\dim(U\cap W)$$ Hint: Show that the map given by $L:U×W\to V$ given by $L(u,w)=u-w$ is linear. I can show that $L:U×W\to V$ given by $L(u,w)=u-w$ is a linear…
hasExams
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Adjoint Operators and Inner Product Spaces

My linear algebra textbook gives the definition of the Adjoint Operator and then says, You should verify the following properties: Additivity: $(S + T)^* = S^* + T^*$ Conjugate homogeneity: $(aT)^* = \overline{a}\,T^*$ Adjoint of adjoint:…
Megan
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Eigenvalues and column space, nullspace

Is there a way to relate Eigenvalues to the column space and nullspace of a matrix? I believe a matrices with different eigenvalues would have a different column spaces and/or nullspace. Is this correct? I am wondering if you can prove that the…
zaz
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