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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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Subspace of polynomials vanishing to order $n$

Let $V\subset\mathbb{R}[x]$ be a vector space of dimension $k$. We say that a polynomial $f$ vanishes to order $n$ at $a\in\mathbb{R}$ if $f(a)=0$ and $n$ is the smallest positive integer such that $f^{(n)}(a)\not=0$. a) Show that $V_n=\{f\in…
morrowmh
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Geometric Interpretation of Matrix Additiom

Is there a geometric meaning to matrix addition similar to how matrix multiplication acts as a linear transformation? I'm really curious thanks!
xEthereal
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Can every real vector space be turned into a Euclidean Vector Space?

Here's what I'm reading right now: So, the question that I have relating to this is if it needs to be proven that an inner product can be defined on every real vector space and if there are infinitely many inner products that can be defined on each…
Abhijeet Vats
  • 6,268
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Finding eigenvalues and eigenvectors for the polynomial equation

I am struggling to find the eigenvectors and eigenvalues for $T: \mathcal{P}(\mathbb{R}) \rightarrow \mathcal{P}(\mathbb{R})$ defined by $Tp=p+p'$ I started by equating $\lambda a_0+\dots+\lambda…
user764658
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3 answers

Proving that a product of certain matrices is not identity

I have a very quick question regarding matrices. Consider $ x= \left( \begin{array}{cc} 1 & 2 \\ 0 & 1 \\ \end{array} \right) $ and $ y = \left( \begin{array}{cc} 1 & 0 \\ 2 & 1 \\ \end{array} \right)$ Clearly no power except $0$ of $x$ or…
ziangzao
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No invertible matrix $P$ such that $PAP^{-1}$ is a diagonal matrix.

Consider a $n \times n$ matrix $A=(a_{ij})$ with $a_{12}=1$, $a_{ij} =0 \ \forall \ (i,j) \neq (1,2)$. Prove that there is no invertible matrix $P$ such that $PAP^{-1}$ is a diagonal matrix. Any help?
mmt
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What is the meaning of $A^TA$?

What does it mean if a matrix is multiplied by its transpose? Informally, it seems like $A^TA$ boils a matrix down to its essentials, but can this operation somehow be understood "intuitively" (e.g. through a geometric interpretation)?
koletenbert
  • 3,970
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Is it possible to diagonalize a matrix without eigenvalues?

Is it possible to diagonalize a matrix without eigenvalues, or in other words, does the diagonal matrix of a diagonalization always have the eigenvalues of a matrix? for example: if a 3x3 matrix has eigenvalues 1,2,3 can you diagonalize that matrix…
user753933
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How to get the eigenvalue of this cases

I'm in a Linear Algebra class, where we are currently covering eigenvalues and eigenvectors. My question is if my answer is correct. Consider the matrix \begin{equation*} A := \begin{bmatrix} 0 & 0 & a \\ b & c & 10 \\ 0 & 0 & a …
Vicente Matus
  • 43
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Existence of a set of matrices

Let $m,n\in \mathbb{N},\ m>n\ge 1$ and the matrices $A_1,A_2,...,A_m\in \mathcal{M}_n(\mathbb{C})$ such that the matrix $\sum_{i=1}^mA_i$ is nonsingular. Prove that there is a set $S\subset \{1,2,...,m\}$ with at most $n$ elements, such that…
BlueSyrup
  • 235
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Rank of $X\otimes Y-Y\otimes X$

Let $X$ and $Y$ be known linear applications $\mathbb{R}^n\rightarrow\mathbb{R}^n$. What can we say about $\mathrm{rank}(A=(X\otimes Y-Y\otimes X))$?
anonymous67
  • 3,458
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Linear algebra skew symmetric

Let A be $3\times3$ matrix such that $Au$ and $u$ are orthogonal for each column vector $u\in\Bbb R^3$. Prove that $A^T=-A$. I thought $(A+A^T)u\cdot u=0$ but it doesn't imply $A^T=-A$, so I'm stuck... Edit : Does this property also hold when we…
Hypernova
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Action of angle-preserving linear transformation on basis vectors

Call a linear map $T: \mathbb{R}^n \to \mathbb{R}^n$ angle preserving in case $T$ is injective and the angle between $Tx$ and $Ty$ is equal to the angle between $x$ and $y$ for all $x, y \in \mathbb{R}^n - \{ 0 \}$, where the angle between $x$ and…
Mr. Chip
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Matrix with each cell equal to sum of its adjacent cells

Consider a $m \times n$ matrix $M$ such that each cell in $M$ is equal to the sum of its adjacent cells (sharing either an edge or a corner with this cell). What are the values in this matrix. I am trying to find a non-zero matrix satisfying this…
Siddharth Joshi
  • 1,488
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How to solve a system of matrix equations involving commutator?

I want to solve the following system of equations: $$XY-YX=-aY+bZ$$ $$ZX-XZ=cY+aZ$$ $$YZ-ZY=dX$$ Where $X,Y,Z$ are all real $3\times3$ matrices which are all different and not a scalar multiple of eachother. $a,b,c,d$ are given and take the values…
Matthew
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