Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, Hamel basis, dimension, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, etc. For questions specifically concerning ...
1
vote
2answers
34 views
Block identity theorem?
If $C$ and $C'$ represent the matrix of a linear transformation with respect distinct bases then we know $C$ and $C'$ are equivalent; in particular we can find invertible matrices $A$ and $B$ such ...
0
votes
1answer
36 views
Number of eigen values of an $N \times N$ matrix
What are the number of eigen values of a non-singular matrix and why ?
What would have happen if matrix is singular, lets say have 2 linear dependent rows.
Preparing for interview, stuck on this ...
2
votes
1answer
24 views
Minimal polynomial matrix
I want to show that $ x^n-1$ is the minimal polynomial of the permutation matrix $P:=(e_2,e_3,....,e_n,e_1)$ where $e_i$ is the i-th unit vector written as a column vector.
And now I have to show ...
1
vote
2answers
34 views
Solving System of Differential equations
The general solution to differential equation
$$x'=Ax$$ where A is a square matrix is given by solving for the eigenvalues and then eigen vectors of matrix $A$. However, is there a general method if I ...
3
votes
2answers
51 views
Linear independence and pairwise angles
Given $p+1$ vectors with pairwise angles $> \pi/2$ in $\mathbb{R}^n$ then every $p$ of them are linearly independent. How to prove this fact?
The motivation for this question is the idea of ...
1
vote
1answer
32 views
The relation between rational forms and jordan forms.
Is there any algorithm / method that allows one to determine the Jordan form of a matrix after determining its rational form?
3
votes
0answers
44 views
Eigenvalues of a tridiagonal trigonometric matrix
Let $A$ be the diagonal matrix w/alternating in sign diagonal entries:
$$ A =
\begin{pmatrix}
\pm \tan(\frac{\pi}{2n+1}) & 0 & 0 & \ldots & 0 \\
0 & \mp ...
2
votes
1answer
35 views
inequality applied to Matrix possible?
My question is this : when is it possible to apply (if at all) a polinomial inequality like this little inequality conjecture ,for example, to a $n\times n$ Matrix $A$ (change the variable $x$ with ...
0
votes
0answers
15 views
Shear decomposition
Is there an algorithm for decomposing a square matrix (or a similar matrix to it) in to shear and diagonal matrices?
All the usual decompositions (Schur, SVD, QR, LU, etc.) don't seem to help. ...
3
votes
3answers
82 views
A is a matrix of integers , prove that A+I is invertible
Question:
$2 \le d \in \Bbb Z$
Let $A \in M_n(\Bbb Q) s.t$ All of it's elements are integers divisible by d.
Prove that $I+A$ is invertible.
What I thought:
I thought of using the determinant of ...
2
votes
1answer
32 views
Matrices manipulation
I am having difficulty with the following question
I have to determine if the following claim is true or not.
If it is true I have to proof it else I need to give an example
I believe it is not ...
0
votes
1answer
31 views
Can the second term of the Schur complement of a symmetric matrix be undefined?
Given the next symmetric matrix conformably partitioned
$$\begin{bmatrix}
A &B \\ B^T &C
\end{bmatrix}$$
I know that $A$ and $C$ are positive definite matrices.
The Schur complement is ...
2
votes
2answers
23 views
Calculating the centralizer of a matrix in a general linear group.
Let $G = GL(3,\mathbb{R})$ be the general linear group over the reals , of order $3$ , and let
$A\in G$ be :
$$
A=\begin{pmatrix}
-1 & 0 & 0 \\\
0 & 1 & 0 \\\
0 & 0 & 2
...
1
vote
1answer
21 views
How to show that every complex matrix with orthonormal columns can be supplemented into an unitary matrix?
Show that every matrix $A \in M_{n,k}(\mathbb{C})$ whose columns are orthonormal vectors in $M_{n1}(\mathbb{C})$ can be supplemented with additional n-k columns to an unitary matrix $U \in ...
2
votes
1answer
34 views
Why $\operatorname{rank}(A^* A)=\operatorname{rank}(A)$ is equivalent to $A^* Ax=0$ if and only if $Ax=0$?
Let $A \in M_{m\times n}(F)$ and $x \in F^n$.
$A^*$ is the adjoint of $A$.
Why is $\operatorname{rank}(A^* A)=\operatorname{rank}(A)$ equivalent to $A^* Ax=0$ if and only if $Ax=0$?
0
votes
0answers
35 views
Doubt on solving the following system with Gaussian Elimination
How to solve tihs problem ?
please send your answer to https://www.facebook.com/dereenjalal.askary?ref=tn_tnmn
or (pianist.81@hotmail.com)
Use Gauss Elimination to solve the following system of ...
2
votes
3answers
66 views
Show that a vector that is orthogonal to every other vector is the zero vector
I have the following question, and I'd like to get some tips on how to write the proof. I know why it is, but I'm still not so great at writing it mathematically.
If $u$ is a vector in ...
0
votes
0answers
21 views
GCD of polynomial in GF(2) and the reals
we were asked to calculate the gcd of $p=x^5+x^4+x^3+x^2+x+1$ and $q=x^4+x^3+x^2+x$ in the fields $\mathbb{R}$ and $GF(2)$
I first did $\frac{p}{q}=x$ with remainder $x+1$
then I did ...
0
votes
2answers
49 views
$Z(GL_n(\mathbb R)) = \{aI : a\neq0\} $ [duplicate]
$Z(GL_n(\mathbb R)) = \{aI : a\neq0\} $
This article is the general case for $GL(n,k)$ where $k$ is a field. Could I prove it only with a basic linear algebra?
1
vote
1answer
18 views
Paralellogram based pyramid volume
Given three points of a paralellogram being
$
P(2,3,4) ; Q(3,1,4); R(2,5,3)
$
I've already calculated the fourth point with
$S(1,7,3)$
Further there's the tip point $Z(5,7,8)$
To get the volume of ...
2
votes
1answer
70 views
How prove that $\;(1-Tr(A))^2+\sum_{1\le i\le j\le 3}(a_{ij}-a_{ji})^2=4\;\;?$
Let $A=\begin{bmatrix}
a_{11}&a_{12}&a_{13}\\
a_{21}&a_{22}&a_{23}\\
a_{31}&a_{32}&a_{33}
\end{bmatrix}$ be an orthogonal matrix with $a_{i,j}\in \mathbb R$, where $\det(A)=1$
...
0
votes
2answers
37 views
Why to obtain the coordinates of vectors in the basis that themselves belong?
Let $ \space T: \mathbb{R^2} \to \mathbb{R^3}$ a linear transformation defined as $ \space T(x,y)=(3x+2y,x+y,-2x-y)$, where $\beta=\{(1,-1),(0,1)\}$ is a basis of $\mathbb{R^2}$. Is not specify a ...
0
votes
1answer
25 views
Extension of the field of scalars.
I want to make sure that I understand this correctly:
$V- \mathbb{Q}$ vector space.
Then $V_\mathbb{R} := V \otimes \mathbb{R} $
is naturally an R-vector space (next to
being a $\mathbb{Q}$-vector ...
0
votes
0answers
23 views
Continuity of a certain matrix-like function
Let $X$ be a finite set and let $M$ be a space of all probability measures on $X$. Let $f:X\to\Bbb R^{m\times m}$ be a random matrix and consider a function $c:M\to\Bbb R$ defined as
$$
c(\mu) := ...
0
votes
1answer
19 views
Vector space: $\forall a\in K, v \in E ( a \cdot_E k=0_E \to a=0_K \lor v=0_E)$
I need to prove the following:
let $E$ vector space on $K$, then $\forall a\in K, v \in E ( a \cdot_E k=0_E \to a=0_K \lor v=0_E)$
Thanks in advance!
5
votes
2answers
23 views
If three corners of a parallelogram are known solve for the 3 possible 4th corners.
An example would be three corners being the points: (1,1), (4,2) and (1,3). I understand the specific solution for this example: (4,4), (4,0) or (-2,2). Which I reasoned when i drew it out. The ...
1
vote
1answer
30 views
Compute $v,W,k$ such that the following is true
$$
\left\{ x \in \mathbb{Z}^4 |
\begin{pmatrix}
5 & 3 & 7 & 0 \\
2 & -4 & 6 & 5
\end{pmatrix}
x =
\begin{pmatrix}
5 \\0
\end{pmatrix}
\right\} = \left\{ v + Wy \ | \ y \in ...
2
votes
0answers
30 views
decomposition according to embeddings
Let $V$ a finite dimensional vector space over $\mathbb{Q}$ and let $F$ be a number field. Assume that there is an injective morphism of rings $F \hookrightarrow End(V)$. I would like to understand ...
0
votes
1answer
33 views
How do I prove $\dim{U} = \dim{W}$ when…
$\mathbf{U} = sp\{(a_1 a_2 a_3),(b_1 b_2 b_3),(c_1 c_2 c_3)\}$
$\mathbf{W} = sp\{(a_1 b_1 c_1),(a_2 b_2 c_2),(a_3 b_3 c_3)\}$
$\mathbf{U}$ and $\mathbf{W}$ are subspaces of $\mathbb{R}^3$
2
votes
1answer
50 views
Continuity of an $\mathbb {R}^2$ function
Let $f$ be an $\mathbb{R}^2$ endomorphism and $N:\mathbb{R}^2\to\mathbb{R}^+$
defined by $$\forall u \in \mathbb {R }^2, N(u) = ||f(u)|| $$
I need to show $N$ is continuous.
The problem is that $N$ ...
8
votes
3answers
60 views
Prove that matrix is non-negative
Problem:
Given $A_{1}, A_{2}, ..., A_{n}$ - finite sets and $a_{ij} = |A_{i}\cap A_{j}|$ - number of elements in intersection of sets. Prove, that matrix $(a_{ij})_{i=1,2,..,n}^{j=1,2,.., n}$ is ...
-1
votes
1answer
68 views
Linear algebra classic, Farkas lemma application
$A \in M_{m \times n}(\mathbb{R})$ and $b \in \mathbb{R}^m$.
Farkas' lemma says exactly one of the following holds:
(a) there exists some $x \in \mathbb{R}^n$, $x \geq 0$, such that $Ax = b$
(b) ...
0
votes
1answer
32 views
Cross product as factor in dot product [duplicate]
Given there are two vectors $w,v$ with $||w||=4$ , $||v||=1$ and $\phi=\frac{2\pi}{3}$
How do you transform the following expression into a form in which it can be computed with the given ...
0
votes
0answers
15 views
conjugate gradient with a positive semidefinite matrix
I'm working on conjugate gradient to solve $Ax=b$ when $A$ is symmetric and positive semidefinite.
When $A$ is symmetric and positive semidefinite, is $(A+\lambda I)$, where $\lambda$ is positive and ...
0
votes
0answers
34 views
Do you have a good source for the determinant of the multilinear operator?
I want to calculate the determinant of the multilinear operator. Do you have a good source for my questions that could help me? Thanks.
4
votes
3answers
30 views
About restriction of linear map
Let $A$ be a linear map from the vector space $X$ to $Y$ and $T$ be a subspace of $X$ . I want to understand what is the meaning of saying that the restriction $A_{|T}: \to A(T)$ is invertible. Could ...
2
votes
1answer
46 views
For a given real square matrix $A$ what is meant by $e^{kA}$ where $k$ is real.
For a given real square matrix $A$ what is meant by $e^{kA}$ where $k$ is real?
I've problem involving this notion and I wondered if $e^{kA}=(e^{ka_{ij}})$ where $A=(a_{ij}).$
0
votes
1answer
44 views
Is't true that for a linear transformation $T:\mathbb R^n\to\mathbb R^n,~T$ is positive definite $\iff\langle Tx,x\rangle>0~\forall~x\ne 0$
Is't true that for a linear transformation $T:\mathbb R^n\to\mathbb R^n,$
$T$ is positive definite $\iff\langle Tx,x\rangle>0~\forall~x\ne 0$
$T$ is negetive definite $\iff\langle ...
2
votes
0answers
43 views
Proof of Sum, Difference, Scalar Multiple of Diagonal Matrices
Assumming A and B are diagonal matrices of the same size, please prove that the following are diagonal matrices as well.
a) $A+B$
b) $A-B$
c) $kA$ , for a scalar $k$
It's not homework- just a ...
2
votes
1answer
38 views
Eigenvectors basis and orthonormal basis for of linear transformation $T$
Is the following true:
Let $V$ be an inner product space, and $T:V\to V $ is a linear transformation, suppose that we have eigenvectors basis of $T$. Thus, there exist orthonormal eigenvectors basis ...
2
votes
2answers
60 views
Infinite dimensional vector space, and infinite dimensional subspaces.
I'm having trouble with this question:
Let $V$ be an infinite dimensional vector space.
Prove that there exist subspaces $U_1,U_2,\dots$ of $V$ with $U_{n+1}\neq U_n$ for all $n\in\mathbb{N}$, such ...
2
votes
1answer
33 views
True or false? About unitary operation.
Let $V$ be a finite inner product space. Let $T:V\to V$ be a linear transformation. Suppose that $v_1,...,v_n$ is an orthonormal basis of $V$ such that $(Tv_i,Tv_i)=1$ for every $1\leq i\leq n$. Then ...
1
vote
1answer
36 views
$(P\Lambda P^{-1}=T^2)~\implies~(\exists \Lambda'~\text{s.t.}~T=R\Lambda' R^{-1})$: $\;P,R\;$ Unitary Matrices
Let $T$ be a linear operator such that the operator $T^2$ is diagonalizable. Is $T$ necessarily diagonalizable?
1
vote
0answers
31 views
Basis of kernel and image of a linear transformation - verification
The transformation matrix I found is: $$\begin{pmatrix} 1 & -1 \\ 1 & 1 \\ 0 & 0\end{pmatrix}$$
Is this how a basis for $\ker$ and $\mathrm{im}$ is calculated?
$$\begin{pmatrix} 1 & ...
3
votes
1answer
70 views
$V=W_1\oplus\cdots\oplus W_k \iff \dim(V)=\sum{\dim(W_i)}$
If $W_1,\dots, W_k$ are subspaces of a finite dimensional vector space $V$ such that $W_1+\cdots+W_k=V$, and I want to show that $V=W_1\oplus\cdots\oplus W_k$ if and only if $\dim(V)=\sum{W_i}$, then ...
0
votes
2answers
45 views
Find the relation between the dimension of the nullspace of $A$ and $A^t$
Let $A$ be a $n \times n$ matrix, what is the relation between the dimension of the nullspace of the homogeneous system of $A$ and the one of $A^t$?
2
votes
0answers
23 views
How to compress a linear operator and have the lossless composition property.
Consider a linear operator on $\mathbf{R}^n$ represented by a square matrix of size $n \times n$, call it $A$. The matrix acts on a row vector, call it $x$ and returns a row vector, call it $x'$, so ...
0
votes
0answers
21 views
Total differential of a map between SU(N) and its quotient group w.r.t. the diagonal subgroup
I am trying to figure out the following statement. Consider SU(N) and its normal subgroup T consisting of diagonal matrices. Then define the map
$\pi : SU(N)/T \times T \rightarrow SU(N) : (xT, t) ...
1
vote
1answer
37 views
Is restriction of scalars well-defined on subspaces?
Let $K/k$ be an extension of fields and let $v_1,\ldots,v_r,u_1,\ldots,u_r\in k^n$. If the span of the $v$'s over $K$ equals the span of the $u$'s over $K$, must the two spans also be equal over $k$?
...
0
votes
1answer
30 views
Proof of cauchy schwarz inequality in inner product space
$0 \le \lVert x-cy \rVert ^2= \langle x-cy,s-cy \rangle = \langle x,x\rangle -\bar{c}\langle x,y\rangle-c\langle y,x\rangle +c\bar{c}\langle y,y\rangle$.
If we set $c=\frac{\langle x,y\rangle}{\langle ...
