Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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3
votes
2answers
59 views

$x=x^2$ in a sub group?

I have a set E defined in ℝXℝ (E=ℝXℝ) and the operation * defined like this ...
0
votes
1answer
10 views

Issues with a particular bilinear form and determining rank, signature, etc. of its restriction

Let $b: M_2(\mathbb{R}) \times M_2(\mathbb{R}) \to \mathbb{R}$ such that $b(X,Y)=trace(X^tAY)$, where $X^t$ is the transpose of $X$ and $A=\begin{pmatrix} 2 & 1\\1 & 0\end{pmatrix}$. In my ...
0
votes
1answer
17 views

Derivative over scalar field with respect to fixed point proof.

Prove there is no such scalar field that $f '(a;y) >0$ for fixed point $a$ and every non-zero vector $y$. I posted this question but some of you pointed out that it is not clear. So, $f ' (a;y)$ ...
1
vote
2answers
28 views

Proving linear dependency for two vector groups

The question: Let V be a vector space over $\mathbb{R}$. Let $S = \{v,u,w\}$ be a group of 3 vectors in V. Let T be defined as $T = \{v, v + u, v + u + 2w \}$. Prove that if S is linearly dependent, ...
0
votes
1answer
24 views

Eigenvector Problem

Given a matrix $X$, let $eigvec(X)$ be its eigenvector associated with the largest eigenvalue. Is there a relationship among $eigvec(X+X^T)$, $eigvec(X)$ and $eigvec(X^T)$? In other words, can I use ...
10
votes
5answers
180 views

Invertibility of a Kronecker Product

Prove that $A\otimes B$ is invertible if and only if $B\otimes A$ is invertible. I don't have a clue where to start to be honest. I am not very familiar yet to the Kronecker Product so could you ...
1
vote
1answer
12 views

Finding the corresponding Perron eigenvalue

Find the Perron root and the corresponding Perron eigenvector of A. $\begin{bmatrix} 0 &1 &1 \\ 1&0&1 \\ 1&1&0 \end{bmatrix}$ I figured out the Perron root which happens to ...
1
vote
1answer
18 views

nullity and rank of the linear transformation $T: T [ p (x)]= p(x+1)$

Let $V$ be the linear space of all polynomials $p(x)$ of degree $\le n$. if $p$ belongs to $V$ and $q = T(p)$, means that $q(x) = p(x+1)$ for all real $x$. find nullity and rank of the linear ...
1
vote
2answers
15 views

Find the matrix $M$, given four vectors

If you have $4$ vectors in a plane $x_1, x_2, b_1, b_2$, and a matrix $M$ such that $Mx_1 = b_1$ and $Mx_2 = b_2$, how do you find $M$ from this given data? Any hints would be appreciated; I am not ...
6
votes
3answers
308 views

Repertoire method for solving recursions

I am trying to solve this four parameter recurrence from exercise 1.16 in Concrete Mathematics: \[ g(1)=\alpha \] \[ g(2n+j)=3g(n)+\gamma n+\beta_j \] \[ \mbox{for}\ j=0,1\ \mbox{and}\ n\geq1 \] I ...
-3
votes
0answers
50 views

Give me an idea

We've got an infinite number of cards, each of them having a positive integer written on it.Prove that however we choose 2015 cards, having the sum of the numbers written on them 4028, we can divide ...
0
votes
1answer
30 views

quotients and direct sums

Let $H$, $K$, $W$, be submodules of a module $M$. Is it true that $(H \oplus K)/W \cong H/W \oplus K \cong H \oplus K/W$? The first seems to follow from 1st isomorphism theorem on the map $\phi = ...
1
vote
1answer
24 views

Nullity and rank of the linear transformation $T[f(t)] = \int_a^b f(t) \sin (x-t) dt ~\forall~x \in [a,b]$

Let $V$ be the linear space of all real functions continuous on $[a, b]$. If $f\in V, g=T(f)$ means that $$g(x)=\int_a^b f(t)\sin(x-t)\,dt\hspace{1 cm} for\ a\le x\le b$$ Then, the nullity and rank ...
1
vote
0answers
23 views

Algebraic multiplicity of an eigen value

Let $T$ be an operator on a complex Vector space $V$. Then, the algebraic multiplicity of an eigen value is equal to $\dim ~null~ (T - \lambda I)^{\dim V}$ Which means, if we obtain the upper ...
0
votes
1answer
27 views

How to calculate time-of-flight and target hit point of a ball thrown against a wall?

Imagine you are throwing a ball against a distant wall, the question is how to find the time taken by the ball to reach the wall and also the point of impact on the wall (after the ball has bounced ...
2
votes
1answer
19 views

The group action of $S_n$ given a partition of $n$

We know that irreducible representations of $S_n$ are given by partitions of $n$. I would like to know if there is a way to explicitly write down the action of some $g \in S_n$ on the representation ...
0
votes
0answers
20 views

Linear maps and subspaces

The set-up for my question is this, let $k \le n$, let $E \subseteq \mathbf{R}^n$ be a $k$-dimensional subspace. Let $I \subseteq \{1,\ldots, n\}$ such that $|I| = k$, then we can define coordinate ...
2
votes
1answer
22 views

extended PCA (tangled matrices)

Given an $m$ by $n$ matrix $A$ and the constant $r$, the principal component analysis allows us to find matrices $W$ and $H$ so that the $WH$ gives a lower rank approximation of $A$. In other words, ...
0
votes
1answer
17 views

Adjoint Transformations and Self-Adjoint Operators

I don't quite understand the whole adjoint and self adjoint thing. I know their definitions: Given a linear transformation $A:\mathbb{R}^d \to \mathbb{R}^m$, its adjoint >transformation, ...
3
votes
4answers
177 views

Identify a formula for each entry of the matrix $\small \begin{pmatrix} 1 & 3 \\ 2 & 6 \end{pmatrix}^n$

Identify a formula for each entry of the matrix $\begin{pmatrix} 1 & 3 \\ 2 & 6 \end{pmatrix}^n$. It's easy to find a solution by just looking at the first few results: \begin{pmatrix} ...
0
votes
1answer
14 views

Inversing badly-conditioned square matrix: methodology

I have a badly-conditioned square matrix. I need to inverse it. For inversing, currently I'm doing the following steps: I take the badly-conditioned matrix with size of $n$ by $n$ By reduced row ...
0
votes
1answer
13 views

Characteristic polynomial of a linear endomorphism of dimension $n$.

So, if $T: V \rightarrow V$ and I suppose that $T^2-3T+2I=0$ and that the $rank(I-T)=k$, what would be the characteristic polynomial of T? I know from previous questions that the eigenvalues of $T$ ...
0
votes
0answers
12 views

Matrix function to express pair-wise distances of rows in $X, Y$

There are two real matrices: $X, Y$ with $X$ being of dimension $n_1$ x $p$, $Y$ of dimension $n_2$ x $p$. The goal is to form the matrix $D$ of dimension $n_1$ x $n_2$ where each element $d_{ij}$ ...
1
vote
1answer
10 views

Projective transformation

I need to find the function $f$ that satisfies the following: $f((1:1:0))=(0:1:1)$ $f((0:1:1))=(1:0:1)$ $f((1:0:1))=(1:1:0)$ If I let: $x=(1:1:0)$ $y=(0:1:1)$ $z=(1:0:1)$, then I get $f(x)=y$, ...
0
votes
1answer
23 views

matrix vs vector span {} linear algebra

I am in a University Linear Algebra course and am confused by the term span and its relation to both matrices and vectors. Can someone help clarify what they mean? =Span= Can it only be made of ...
0
votes
3answers
35 views

Sufficient and necessary conditions to obtain a solution

Find sufficient and necessary conditions for which the following system of equations: $$ax+by=c$$ $$dx+sy=h$$ $$qx+wy=v$$ has at least one real solution $(x,y)$. Here $a,b,c,d,s,h,q,w,v$ are real ...
1
vote
2answers
33 views

General form of an element of the othogonal basis of $q$

Let $$q \begin{pmatrix}a & b \\ c & d\end{pmatrix}= (a-b)^2+(b-c)^2+(c-d)^2$$ quadratic form on $M_2(\mathbb{R})$. How can I prove that every orthogonal basis $B$ of $M_2(\mathbb{R})$ has ...
0
votes
0answers
14 views

See if vector set is basis of space using Gram Schmidt process

I have a problem my teacher gave me and I can't find an answer. She gave me a set of 3 vectors, $$\begin{bmatrix} 1 \\ 0 \\ 1 \end{bmatrix} \begin{bmatrix} 7\\3\\1 \end{bmatrix} \begin{bmatrix} ...
3
votes
2answers
726 views

About finding $2\times 2$ matrices that are their own inverses

They ask me to find all invertible matrices $A$ of the form: $\begin{bmatrix}a & b\\ c&d \end{bmatrix}$ and satisfying $A=A^{-1}$ and $A^t=A^{-1}$. I find that rather complex; does it have ...
0
votes
2answers
18 views

Let $V$ be a finite dimensional linear space and let $S$ be a subspace of $V$. Prove that a basis for $V$ need not contain a basis for $S$.

Let $V$ be a finite dimensional linear space and let $S$ be a subspace of $V$. Prove that every basis of $S$ is part of a basis for $V$ but a basis for $V$ need not contain a basis for $S$. Attempt: ...
0
votes
1answer
18 views

Prove that the LDU factoriztion is unique [on hold]

How would one prove that the LDU factorization of a matrix is unique?
0
votes
1answer
101 views

change the matrix when we extend the field

Let $M$ be an $F_pC_q$- module represented by the matrix $$\left( \begin{matrix} a & b\\ c & d \end{matrix}\right)$$ i.e., $m_1 g=am_1 + bm_2$ and $m_2g=cm_1 + dm_2$ where g is the ...
-2
votes
0answers
21 views

True or False. The intersection of any two subset of vector $V$ is a subspace of vector $V$. [on hold]

The intersection of any two subsets of a vector space $V$ is a subspace of the vector space $V$. Explain if it is true or false.
2
votes
1answer
86 views

Linear Transformation on $\mathbb{R}^6$

Let $W$ be a vector space over $\mathbb R$ and let $T:\mathbb R^6 \to W$ be a linear transformation such that $S = \{Te_2, Te_4, Te_6\}$ spans $W$. Wich one of the following must be true? ...
0
votes
1answer
15 views

Maximizing the number of zero entries in a linear combination of matrices

I was wondering if there exists an algorithmic way of solving the following problem. Let's say you have a bunch of square $N\times N$ matrices (call them $M_i$), and you want to form a linear ...
0
votes
2answers
51 views

Find a positive definite matrix B such that $B^2=A$. [on hold]

Find a positive definite matrix B such that $B^2=A$, where $$A=\begin{pmatrix} 2&-1\\ -1&2 \end{pmatrix}$$
0
votes
1answer
14 views

How to prove that $B$ is positive definite when $\|A-B\|\leq\lambda_\min(A)$ for some positive definite $A$?

Denote by $\mathbb R^{n \times n}$ the vector space of $n \times n$ matrices with real entries. For $A \in \mathbb R^{n \times n}$, the notation $A\succ 0$ means that $A$ is symmetric and positive ...
-1
votes
0answers
31 views

Please guide me what are the topics i need to study in maths from basic. [on hold]

I am not having good knowledge in maths.Please guide me what are the topics i.e (algebra,calculus,diff.eqn...)i need to study by step by step. please guide me.
1
vote
0answers
47 views

How to prove that the limit of solution is $-\pi^2/3$?

Consider a matrix $M$ with $2N+1$ rows and columns, so that for $p=0,1,...,2N$ and $k=-N,...,N$ matrix elements ($k$ indexes columns, $p$ is index of row) are $$M_{pk}=\frac{k^p}{p!}.$$ Taking a ...
0
votes
1answer
17 views

Injective acton of an Chevalley generator of $\mathfrak{sl}(2)$ on non-intergral weight module

I have a problem as follows. Let $E_{i,j}\in M_2(\mathbb{C})$ be the elementary matrix and $\mathfrak{g}:=\mathfrak{sl}(2) = \{ A \in M_2({\mathbb{C}})| tr(A) =0 \}$ the special linear Lie algebra. ...
1
vote
2answers
543 views

Gauss-Seidel method convergence algorithm

From Wikipedia: The convergence properties of the Gauss–Seidel method are dependent on the matrix A. Namely, the procedure is known to converge if either: ...
4
votes
2answers
260 views

Why does Gaussian elimination not preserve similarity of a matrix?

I am trying to understand reduction of an unsymmetric real square matrix to Hessenberg form from Numerical Recipes Vol. 3. In it, the author states that one does not use Gaussian elimination for ...
0
votes
1answer
38 views

Proof of dimension equality

Is my proof of $\mathrm{dim}(U+W)=\mathrm{dim}(U)+\mathrm{dim}(W)-\mathrm{dim}(U \cap W )$ correct?.Suppose we the basis of $U \cap W$ is $B_{0}$ then we can add this basis to the basis of W let ...
1
vote
2answers
24 views

Linear transformation Df=$\frac{df}{dx}$

Let $Rx$ define vector space of all real polynomials. Let $D:Rx \to Rx$ denote map Df=$\frac{df}{dx}$, for every f. Then which of following is true. $D$ is one-to-one $D$ is onto There exist $E:Rx ...
0
votes
2answers
340 views

Given a $4\times 4$ symmetric matrix, is there an efficient way to find its eigenvalues and diagonalize it?

I have a $4\times 4$ matrix $$A=\left(\begin{array}{cccc}8 & 11 & 4 & 3\\11 & 12 & 4 & 7\\4 & 4 & 7 & 12\\3 & 7 & 12 & 17\end{array}\right).$$ I want to ...
0
votes
0answers
9 views

Blocks in a layer? $X$ layers of blocks in a triangle, which $Y$ being the total number of blocks… (w/o using triangular numbers) [on hold]

I have $32$ layers of blocks in a triangle. However, for the sake of variation, let's call that $X$. I have $1000$ square blocks total. We'll call those $Y$. The first layer of the triangle has $1$ ...
0
votes
0answers
25 views

Proof the $\mathrm{rank(rows)=rank(columns)}$

Assume we have matrices $A=BC$.It is obvious that the $i$th row of $A$ is a linear combination of the rows of $C$ with coefficients from the $i$th row of $B$ or $b_{i1}C_1........b_{in}C_n$. ...
2
votes
1answer
714 views

The form of $2 \times 2$ unitary matrices

I've been working through "Groups and Symmetry" (Armstrong) and came across this problem in chapter 9 which I can't figure out. Any hints/help would be greatly appreciated! Show that every $2\times ...
3
votes
2answers
53 views

Skew symmetric $4\times 4$ matrix of full-rank

I have come across the fact that a $4\times 4$ skew-symmmetric matrix of full-rank is equivalent to \begin{pmatrix} 0 &\theta_1& 0 &0 \\ -\theta_1& 0 &0 &0 \\ 0& 0&0 ...
1
vote
1answer
391 views

Largest eigenvalue, semidefinite programming

The problem is to minimize the largest eigenvalue of a function of $x$. objective: $$ \min\{\lambda_{\max}(A(x))\}$$ where $$A(x) = A_0+x_1A_1+x_2A_2+...x_nA_n$$ and all $A$ is positive semidefinite. ...