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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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Equivalence between two matrix expressions

Let $G$ be a $n \times n$ symmetric and invertible matrix, and $b \in \mathbb{R}^n$, and let $V$ be a full-rank $n \times (n-1) $ matrix, such that $b^T V = 0 $, i.e. the vector $b$ is orthogonal to all the columns of $V$. The first expression is $…
Hosam Hajeer
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Prove or disprove: If $u,v \in \Bbb R^n$ are linearly independent column vectors, then $\operatorname{rank}(uv^T-vu^T)=2$

Prove or disprove: If $u,v \in \Bbb R^n$ are linearly independent column vectors, then $\operatorname{rank}(uv^T-vu^T)=2$. I can see that $\operatorname{rank}(uv^T-vu^T) \leq 2$ since $\operatorname{rank}(A+B) \leq…
Messi Lio
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Why is the derivative map a linear transformation?

My textbook defines a linear transformation as a linear map from a space into itself $t:V \rightarrow V$, so basically where the domain equals the codomain. It then goes on to say that the derivative map d/dx: $P_n \rightarrow P_n$ is a linear…
Mezzoforte
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The possible set of eigenvalues of a $4\times 4$ skew symmetric, orthogonal matrix

The possible set of eigenvalues of a $4\times 4$ Real skew symmetric, orthogonal matrix is $1.\{\pm i\}$ $2.\{\pm i,\pm 1\}$ $3.\{\pm 1\}$ $4.\{\pm i,0\}$ As it is real skew symmetric so eigenvalues may be $0$ or Purely Imaginary, and as it is…
Myshkin
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What, along with homogeneity, implies additivity?

Suppose $f$ is a function defined on a vector space which satisfies one of the requirements for a linear function, the “homogeneous” condition: $$\alpha \cdot f(v) = f(\alpha v).$$ This doesn't imply that $f$ must satisfy the other condition of a…
Jbag1212
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Given $A,B$ are $n \times n$ matrices, $(A+B)^2=A+B,r(A+B)=r(A)+r(B)$. Prove: $A^2=A,B^2=B$

Given $A,B$ are $n \times n$ matrices, $(A+B)^2=A+B,r(A+B)=r(A)+r(B)$. Prove: $A^2=A,B^2=B$ I don't know how to use $r(A+B)=r(A)+r(B)$
Doggy
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The adjoint and inner product space

Do these real spectral theorems hold for a general inner product? Version 1: Let $V$ be a finite dimensional inner product space. Suppose that $T\in L(V, V)$ is self-adjoint. Then, there exists an orthonormal basis $B\subset V$ consisting of…
user1104987
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Mark the points $\frac{3}{4}v + \frac{1}{4}w$ from $\frac{1}{2}v + \frac{1}{2}w$

Recently, I do self-learning "Linear Algebra" by using this book "Introduction to Linear Algebra, 3rd Edition" by Gilbert Strang with his lecture on MIT Opencourseware. I am having problem with one of his problem in his book. The following is the…
Monkey D Luffy
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Where do 4 planes in 4D space intersect?

Problem 8 in problem set 2.1 in Gilbert Strang's Linear Algebra says "Normally 4 planes in a 4-dimensional space meet at a ____." Let the four dimensions be $x,y,z,w$. In three dimensions, a plane is defined to be a two dimensional surface that…
Saksham Sethi
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Why is this step in a proof of the Woodbury matrix identity valid?

I'm reading over a proof for the Woodbury matrix identity. However, there is a part of the proof that I'm not sure of (highlighted in red below). The Woodbury matrix identity states that, given the complex matrices $A \in \mathbb C^{n \times n},B…
mhdadk
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Is this isomorphism natural?

Suppose I constructed a linear map $\phi$ without choosing a basis, but in order to check that $\phi$ is an isomorphism, I chose a basis. Is $\phi$ still considered a natural isomorphism? Edit: The problem is asking to construct a map for…
beeflavor
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Isomorphic finite dimensional vector spaces

Theorem: Two finite-dimensional vector spaces over $F$ are isomorphic if and only if they have the same dimension. $F$ denotes either $\mathbb{R}$ or $\mathbb{C}$ Below is proof of the theorem. Where in the proof do we assume that the vector spaces…
jenny9
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Rank and nullity theorem proof question

I was looking at my professor's proof that the dimension of the image is given, following the rank and nullity theorem, by the difference of the dimension of the domain and the dimension of the nullspace. He proves it by showing that if you extend a…
Luca Mautino
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$AB=BA$ if there is an orthonormal basis of $\mathbb{R}^n$ of eigenvectors

Show that if there is an orthonormal basis of $\mathbb{R}^n$ that consists of eigenvectors of both of the $n \times n$ matrices $A$ and $B$, then $AB = BA$. I'm not sure if what I have done suffices to solve the problem, but let…
Sarunas
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If a vector $v$ is an eigenvector of both matrices $A$ and $B$, is $v$ necessarily an eigenvector of $AB$?

I'm preparing for my final and this question came up in one of the practices. I am tempted to say no, but I've been having trouble proving this. If a vector $v$ is an eigenvector of both matrices $A$ and $B$, is $v$ necessarily an eigenvector of…
Joseph
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