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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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Show A and B have a common eigenvalue

Let A, B and C complex square matrices such that: $ C\neq 0 $ and $AC=CB $ prove that A and B has a common eigenvalue. It's worth mentioning that earlier in the assignment I have proved that $A^{n}C=CB^{n}$, but I'm not sure how to use it. This is…
user7610
  • 605
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When a complex matrix is similar to a real matrix?

Suppose I have a matrix whose entries are in $\mathbb{C}$. How easy or hard is it to tell in general if a matrix $M$ is similar to a real matrix? Rasmus pointed out in the comments that in general a matrix has no similar real counterpart. What about…
Mark
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Confusion regarding definition of $F^\infty$ in Sheldon Axler's *Linear Algebra Done Right*

In his linear algebra book, Sheldon Axler defines the set of all sequences of elements of $F$ as: $$F^\infty = \{(x_1, x_2, \ldots): x_j \in F\text{ for } j = 1, 2, \ldots\}.$$ He also says: Sometimes we will use the word list without specifying…
Clojure
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How do you use the Gram-Schmidt process to generate an orthonormal basis of $\mathbb{R}^3$?

These vectors form a basis on $\mathbb R^3$: $$\begin{bmatrix}1\\0\\-1\\\end{bmatrix},\begin{bmatrix}2\\-1\\0\\\end{bmatrix} ,\begin{bmatrix}1\\2\\1\\\end{bmatrix}$$ Can someone show how to use the Gram-Schmidt process to generate an orthonormal…
jmendegan
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R is uncountable as a Q-vector space

I just wanted to ask whether my proof is correct: Suppose instead that $\mathbb{R}$ had a countable $\mathbb{Q}$-basis, say $v_1,v_2,v_3,\ldots$ (possibly finite). Since $\mathbb{Q}$ is countable, $\,\text{span}(v_1,\ldots,v_k)$ is countable for…
user166724
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Orthogonality of the decomposition of a vector space over one of its endomorphisms

Let $V$ be a finite-dimensional real inner product space and let $\tau$ be an endomorphism of $V$. Let $V=V_1 \bigoplus \cdots \bigoplus V_r$ be the decomposition of $V$ into $\tau$-invariant and $\tau$-cyclic subspaces, corresponding to the…
Manos
  • 25,833
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Divide by a vector?

When doing matrix multiplication can I carry a vector to the other side? For example if I have: $Ab = c$ where A is m by m invertable matrix, and b is m by 1 col vector, c m by 1. Can I do something like this: $A = c/b$ And what does that mean... I…
Andriy Drozdyuk
  • 889
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If rank$(A)=r$, show that rank$(A^\top A)=r$

Let $A$ be $m\times n$ matrix with rank $r=\min(m,n)$. How do we show that rank$(A^T A)$ is $r$.
Lionville
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Finding the basis of an intersection of subspaces

We have subspaces in $\mathbb R^4: $ $w_1= \operatorname{sp} \left\{ \begin{pmatrix} 1\\ 1 \\ 0 \\1 \end{pmatrix} , \begin{pmatrix} 1\\ 0 \\ 2 \\0 \end{pmatrix}, \begin{pmatrix} 0\\ 2 \\ 1 \\1 \end{pmatrix} \right\}$, $w_2=…
GinKin
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Can $A, B$ fail to commute if $e^A=e^B=e^{A+B}=id$?

Consider the real $n \times n$-matrices $A$ and $B$. Can $A, B$ fail to commute if $e^A=e^B=e^{A+B}=id$ ?
Siggi
  • 67
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Transpose of a linear operator is well-defined

I would like to define the transpose of a linear operator $T$ between finite-dimensional vector spaces $V$ and $W$. Wikipedia gives a straightforward one, but I would like to define it based on what I already know about matrices. Thus, I define the…
Brian Burns
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Linear Algebra determinant reduction

Prove, without expanding, that \begin{vmatrix} 1 &a &a^2-bc \\ 1 &b &b^2-ca \\ 1 &c &c^2-ab \end{vmatrix} vanishes. Any hints ?
square_one
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Finding vectors in not in union of subspaces

Possible Duplicate: A vector space over $R$ is not a countable union of proper subspaces This is a single step in a larger homework problem that I'm having difficulty with. Consider a finite set of vectors $A$ in $\mathbb{R}^n$ of length $k$ $A$…
user149100
  • 151
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Is the empty set is a subspace of any vector space

Is the empty set is a subspace of any vector space? im not too sure about this one, is the zero vector in the empty set?
user146122
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$W$ is a subspace of $\mathbb{R}^n$ and $K$ is a compact subset of $V$ with $W \cap K = \emptyset$.

Suppose $W$ is a subspace of $\mathbb{R}^n$ and $K$ is a compact subset of $V$ with $W \cap K = \emptyset$. Show that there exists a vector $v \in V$ such that $\langle v,w \rangle = 0$ for all $w\in W$ and $\langle w,x \rangle <0$ for all $x\in…
user53970
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