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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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Prove that a set of matrices is a subspace

I'm self studying linear algebra and now I'm starting with proofs and so on. I found this exercise and this is the way I prove it. I think it's correct but I'm not sure I mean, what do you think? Is the set of matrices $ \begin{pmatrix} x &&…
Susana
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Let $T: \mathbb{R}^3→\mathbb{R}^3$ be a linear transformation. Show that there is a line $L$ such that $T(L) = L$.

Let $T: \mathbb{R}^3→\mathbb{R}^3$ be a linear transformation. Show that there is a line $L$ such that $T(L) = L$. I am totally stuck on it.how can I able to solve this problem?please somebody help.thanks for your kind help.
priti
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Principal angles between subspaces

Let $A$, $B$ be $k$-dimensional subspaces of an Euclidean space $V$ of dimension $\geq 2k$. How to find an orthonormal system $x_1,...,x_{2k}$ in $V$ and numbers $\phi_1,...,\phi_k \in [0,\frac{\pi}{2}]$ such that $\{x_1,...,x_k \} $ is a basis…
Richard
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Find the eigenvalues and eigenvectors of the matrix with all diagonal elements as $d$ and rest $1$

A matrix has all elements 1 except the diagonal elements. It is an $n\times n$ matrix. What are the eigenvectors and eigenvalues ? Solving book problems in Strang book and stuck on this one and I have no idea where to begin ?
user669083
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If the matrix of a linear map is independent from the basis, then the map is a multiple of the identity map.

Let $V$ be a finite dimensional vector space over $F$, and let $$T:V\to V$$ be a linear map. Suppose that given any two bases $B$ and $C$ for $V$, we have that the matrix of $T$ with respect $B$ is equal to that with respect to $C$. How can we show…
Spenser
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Proving that $\|A\|$ is finite.

Let $|v|$ be the Euclidean norm on $\mathbb{R^n} $. For $A\in \mathrm{Mat}_{n\times n}(\mathbb{R})$ we define $\displaystyle \|A\|:= \sup_{\large v\in \mathbb{R^n},\,v \neq 0}\frac{|Av|}{|v|}$. How to show that $\|A\|$ is finite for every $A$? It…
LOLA
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What does the double-lined capital $\mathbb{E}$ (not the sigma) stand for?

I've encountered this symbol that looks like a capital $\mathbb{E}$ (with double vertical lines), which I am not familiar with, and I have no idea what to search for to find what it means, so apologies if it is something trivial. The context in…
jbx
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Reducible and Irreducible polynomials are confusing me

The definition claims that a polynomial in a field of positive degree is a reducible polynomial when it can be written as the product of $2$ polynomials in the field with positive degrees. Other wise it is irreducible. So if a polynomial $f(x)$ can…
ming
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How to calculate basis of kernel?

I have a linear transformation. The transformation and what I tried is written on the attached work page. Is my way wrong? what is the basis of KerT? LinearAlgebra: S -> S is a joke with my friends. sorry for this.
Billie
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are there any "deep" reasons for representing linear systems as $Ax=b$ instead of $xA=b$?

Nowadays we represent the system of $m$ linear equations $$\sum_{i=1}^na_{1i}x_i=y_1$$ $$\sum_{i=1}^na_{2i}x_i=y_2$$ $$\vdots$$ $$\sum_{i=1}^na_{mi}x_i=y_m$$ as $\mathbf{Ax}=\mathbf{y}$, where $(A)_{ij}=a_{ij}$ is an $m\times n$ matrix, $\mathbf{x}$…
symplectomorphic
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Extending a real vector space into a complex vector space using a linear map

Let $V$ be real $n$-dimensional vector space, and $T:V\to V$ is a linear map satisfying the condition $T^2(v)=-v$ for all $v \in V$. Then, Show that $n$ is an even integer. Use $T$ to make $V$ into a complex vector space such that the…
Sayantan
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Is there a scalar product s.t. the following list is orthogonal?

Let $A_1=\begin{pmatrix}1&0\\0&0\end{pmatrix}\quad A_2=\begin{pmatrix}1&1\\0&0\end{pmatrix},\quad A_3=\begin{pmatrix}1&1\\1&0\end{pmatrix},\quad A_4=\begin{pmatrix}1&1\\1&1\end{pmatrix}$. Is there a scalar product s.t. $\|A_k\|=k$ for $k=1,2,3,4$…
MSE
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Why we have introduced linear algebra?

I am new to linear algebra and am trying to find the motivation behind defining it in such a way and need for defining it. To study $2$-D,$3$-D space we have geometry, so why do we need linear algebra then? Why did they choose the exact properties…
user579781
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What is the significance of the order in an ordered basis/basis?

Throughout my Linear Algebra course I heard reference to the fact that a set must be ordered in some way to be a basis for a space, but never managed to see the importance of this - what is it? What would the consequences be of re-ordering our basis…
user27182
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Why $A$ invertible $\iff \det A\neq 0$

Let $A$ a matrix $n\times n$ over $\mathbb R$. I'm trying to prove that $A$ is invertible $\iff\det A\neq 0$. If $A$ is invertible, there is $B$ s.t. $AB=I$, and thus $$\det(A)\det(B)=\det(AB)=\det(I)=1,$$ and thus $\det A\neq 0$. I have problem to…
user330587
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