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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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Steinitz exchange lemma

How can I show this? if $b_{1}, ..., b_{n+1}$ are linears combinations of $a_{1}, ..., a_{n}$ then $b_{1}, ..., b_{n+1}$ are linearly dependents. In my textbook they call it Steinitz lemma. I wonder if is it equivalent…
mohamez
  • 1,521
5
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3 answers

Finding a vector orthogonal to a subspace

Suppose you were given vectors $a_1,\dots,a_n \in \mathbb{R}^m$ then how would you compute some vector orthogonal to the given list of vectors? Note that you are allowed to return the zero vector only if the vectors span $\mathbb{R}^m$. I thought…
anon
  • 51
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6 answers

A matrix of order 8 over $\mathbb{F}_3$

What is an example of an invertible matrix of size 2x2 with coefficients in $\mathbb{F}_3$ that has exact order 8? I have found by computation that the condition that the 8th power of a matrix $\begin{bmatrix}a & b\\c & d\end{bmatrix}$ is the…
user700841
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0 answers

If $\mathbf{R}$ is an upper triangular matrix, then does $\|\mathbf{R}\| \le \|\mathbf{R} + \mathbf{R}^T\|$ hold?

Let $\mathbf{R}\in\mathbb{R}^{n\times n}$ be upper triangular and $\|\cdot\|$ be the induced 2-norm of matrices. Then, does $\|\mathbf{R}\| \le \|\mathbf{R} + \mathbf{R}^T\|$ hold?
flyingwith
  • 872
5
votes
2 answers

Show that the orthogonal projection onto $Range T$ is equal to $T(T^\ast T)^{-1}T^\ast$ given that $T: V \to W$ is injective

Given $V$ and $W$ as finite-dimensional inner product spaces and an injective linear map $T: V \to W$, how can we show that $$P_{Range T} = T(T^\ast T)^{-1}T^* \in \mathcal{L}(W)$$ where $P_{Range T}$ is the orthogonal projection onto $Range T$ and…
allonsy676
  • 157
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4 answers

Prove or disprove: If $x^T A x = 0 $ for all $x$, then $ A = 0 $.

Let $A$ be a square matrix and $x$ be a vector. Now consider the statement: If $x^T A x = 0 $ for any $x$, then $A = 0$. Is the above statement true or false? How would you prove it?
user1769197
  • 1,227
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0 answers

linear algebra : show that $~f~$ is a linear map.

We say that a sequence $(U_n)_{n \in \Bbb N} \subset \mathbb{R}$ is Fibonacci if it satisfies $\ U_{n+2} = U_{n+1} + U_n, \ \forall n \in \mathbb{N}$. Let $F$ be the set of all Fibonacci sequences. We have the function $f: F \to \mathbb{R} \times…
Dhia Othmani
  • 81
5
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3 answers

Two different definitions of vector space

I have two different linear algebra books and realized that the definitions of vector space on them are slightly different. One of the definition has the following statement for the condition of scalar multiplication and the other does not: "For all…
Tengu
  • 1,078
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0 answers

Linear least squares: overdetermined system necessary? and finding solutions?

The wikipedia article on linear least squares only considers overdetermined systems (rows $\geq$ columns). I'm confused if this assumption is really necessary or not. Given any matrix $A, \|Ax - b\|^2$ is convex and differentiable so the minimum is…
student
  • 101
5
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2 answers

Prove $v_1,v_2,\cdots,v_m $ are linear independent.

$\Bbb{E}^n$ is an Euclidean space with dimension $n$. $v_0,v_1,\cdots,v_m\in \Bbb{E}^n,m\le n $ , and $(v_i,v_j)\lt0$ for $0\le i\ne j\le m$. Prove $v_1,v_2,\cdots,v_m $ are linear independent. My try: I tried to make an argument like this, if…
Jaqen Chou
  • 1,057
5
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6 answers

How to find zero of an space

I was trying to solve this exercise: Decide whether or not the set $\Bbb R^2$, with addition defined by $$(x,y) + (a,b) = (x+a+1, y+b)$$ and with scalar multiplication $$r\cdot(x,y) = (rx+r-1, ry)$$ is a (real) vector space. So, I try to prove…
Randolf Rincón Fadul
  • 762
5
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1 answer

let $M$ be a Hermitian matrix of order $n\times n$ with rank $k (\neq n)$

let $M$ be a $n \times n$ matrix of rank $k (\neq n)$ if $\lambda \neq 0$ is an eigenvalue of $M$ with corresponding unit column vector $u$. with $Mu=\lambda u$,then which of the following is\are true?. 1). $rank(M-\lambda uu^{*})=k-1$ 2).…
gaurav saini
  • 763
5
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3 answers

Find $n \times n$ matrices $A$ such that $\det A = 0$ and $\text{rank}(AB) = \text{rank}(BA)$ for any $n \times n$ matrix $B$

Find all complex-valued $n \times n$ matrices $A$ such that $\det A = 0$ and $\text{rank}(AB) = \text{rank}(BA)$ for any $n \times n$ complex-valued $B$. I believe that $A = 0$ is the only answer. I have been able to prove that, if $A$ is of rank…
Tanny Sieben
  • 2,471
5
votes
2 answers

Find a basis $A$ of space $\mathbb R^{4}$ and basis $B$ of space $\mathbb R^{3}$

Let $\varphi: \mathbb R^{4} \rightarrow \mathbb R^{3}$: $$\varphi(x_{1},x_{2},x_{3},x_{4})=(x_{1}+x_{3}+x_{4},x_{1}+x_{2}+2x_{3}+3x_{4},x_{1}-x_{2}-x_{4})$$Find a basis $A$ of space $\mathbb R^{4}$ and basis $B$ of space $\mathbb R^{3}$ such that…
MP3129
  • 3,195
5
votes
1 answer

Exponential of a skew-skymmetric matrix

If $A$ is a skew-symmetrical matrix with it's diagonal elements as $0$, Prove that it's exponent $e^A$ is an orthogonal matrix.
Cyanide2002
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