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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28 views

Matrix with entries equal to $1$ and $-1$ (Sign Matrix)

What can we say about the determinant and (or) maximum eigenvalue of a matrix with entries equal to $1$ and $-1$. Further assume that the rows and columns are linearly independent. Are there special ...
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2answers
24 views

which is the inverse of this linear application?

$L:C[0,1]\rightarrow C[0,1]$ $L[f(x)]=\int_{0}^{x}f(s)ds$ that is linear and R(T)=$C^{1}[0,1]$ s.t $L(0)=0$. I need calculate $L^{-1}: R(T)\rightarrow C[0,1] $ I could give some suggestion ?
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1answer
58 views

Constructing a bases for $U_{1}+U_{2}$

I have the following: Let $L_{1}(x_{1},x_{2},x_{3},x_{4})=(3x_{1}+x_{2}+2x_{3}-x_{4}, 2x_{1}+4x_{2}+5x_{3}-x_{4})$ and $L_{2}(x_{1},x_{2},x_{3},x_{4})=(5x_{1}+7x_{2}+11x_{3}+3_{4}, ...
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0answers
13 views

show the following equivalence

Let $A=(a_{ij}) \in C^{nxn}$ be a self-adjoint matrix such that A*=A.Show that the following are equivalent: A is positive if and only if the determinant of the matrix $A^k$=$$\begin{bmatrix}a_{11} ...
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2answers
30 views

Show $f=f^*$ including inner product

Let $V$ a $\mathbb{C}$-vector space with inner product $\langle \cdot , \cdot \rangle$ and $f:V\to V$. Show that if $\langle f(v),v\rangle\in \mathbb{R}$ for $v \in V$, then $f=f^∗$. I was thinking ...
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2answers
30 views

Number of vectors over a finite field that are linearily independent to a subspace

let $S$ be a vector space over a finite field of size $q$ and let $T$ be a subspace of $S$. I am looking for a formula or an algorithm to compute the number of vectors from $S$ that are independent ...
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2answers
20 views

Grasping “Substitution” in terms of linear algebra

So I have a set of equations: $$x_{1} + x_{2} = 1$$ $$x_{2} + x_{4} = 3$$ From linear algebra, we know that (say, we're in $\mathbb{R}^{4}$, i.e. we have 4 variables), the solution space to the ...
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0answers
13 views

Determinant 1 matrix does not change p-adic measures

Let $f:\mathbb Z^d \rightarrow \mathbb Z^d$ be a linear map having determinant 1. Is there an obvious way to see that if $U\subseteq \mathbb Z_p^d$ is a measurable set, then the p-adic measure of $U$ ...
2
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1answer
27 views

Example for a norm on Hom(V,W) which is not determined by rank-one operators

Assume $(V,\|\cdot \|_V),(W\|\cdot \|_W)$ are two finite dimensional normed spaces (over $\mathbb{R}$). Any operator norm on $\text{Hom}(V,W)$ is determined by its value on rank-one operators. (This ...
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1answer
31 views

Finding a basis for $V, W, V+W$ and $V \cap W$

Problem: Let \begin{align*} V = \left\{(x,y,z,u) \in \mathbb{R}^4 \mid y+z+u = 0 \right\} \end{align*} and \begin{align*} W = \left\{(x,y,z,u) \in \mathbb{R}^4 \mid x+y = 0, z = 2u \right\} ...
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24 views

Meaning of points in 3D space that lie on a line

$\{X_i\}_{i=1}^n$ are $n$ points in a three dimensional space with basis functions $\{\phi_i\}_{i=1}^3$. For simplicity we can assume that this the space is $\mathbb{R}^3$ with the natural basis ...
2
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1answer
19 views

Find the elementary divisors of a matrix given its characteristic and minimal polynomials

This question comes from and old exam: Suppose the square rational matrix $A$ has characteristic and minimum polynomials $p_A(x) = x^6(x^2-2)^3(x^2+4)^2$ and $m_A(x) = x^2(x^2-2)(x^2+4)^2$ and $null A ...
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1answer
11 views

Use of GS before projecting a vector onto a plane

I need help with the following exercise: Given the vectors $u_1 = (2,-1,2), u_2 = (1,2,1), u_3 = (-2,3,3)$, what is the projection of $u_3$ onto the plane spanned by $u_1$ and $u_2$. I'm not sure if ...
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votes
3answers
47 views

Optimal approximation of quadratic form

Let $\mathbf{x}\in\Bbb{R}^n$ and $A\in\Bbb{S}_{++}^n$, where $\Bbb{S}_{++}^n$ denotes the space of symmetric positive definite $n\times n$ real matrices. Also, let $Q\colon\Bbb{R}^n\to\Bbb{R}_{+}$ be ...
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3answers
73 views

Are there nontrivial vector spaces with finitely many elements?

I have only seen infinite vector spaces and the one finite vector space i.e the trivial vector space $\{0\}$. Is there any other finite vector space?
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1answer
52 views

Characteristic polynomial of A, if $\det(\operatorname{adj}(\operatorname{adj}(A))) = 81$?

Let $A$ be a square real matrix whose eigenvalues are positive integers, with $$\det(\operatorname{adj}(\operatorname{adj}(A))) = 81 \, .$$ What is the characteristic polynomial of A? Any hints? ...
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0answers
16 views

Span of a projection matrix $P(\theta, \phi)$

I have a projection matrix which depends on two parameters, $\theta$ and $\phi$. I am interested in finding out if the relationship between space spanned by the projection matrix for say $\theta_1$, ...
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14 views

Why does $M$ have a limit rank in the operator norm? Why is $S$ bounded?

Define operator $S$ and $M$ on $\ell^2$ by $(SX)_n = \begin{cases} 0 & n = 0 \\ x_{n - 1} & n > 1 \end{cases}$ $(Mx)_n = \dfrac{1}{n + 1} x_n,\qquad n \ge 0$ Why does $M$ have a limit ...
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2answers
25 views

Canonical linear mapping is bijective

Let $V$ be a $K$-vector space with finite dimension. Proof that mapping: $V^* \otimes V \rightarrow {\rm End}_K(V), \ h\otimes a\mapsto (x\mapsto h(x)a)$ is bijective. So we have one mapping, which is ...
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1answer
20 views

Calculating Determinant Using an Equation

$detA_{6x6} \neq 0$. $2A+7B=0$ Calculate $6det(2(A^t)^2B^{-1}A^{-1})$ My solution attempt: $A = -7/2*B$ and $det A^t = det A$ so $6det(2*A*(-7/2B)*B^{-1}A^{-1}) = 6det(-7)= 6*(-7)^6 = 705894$ ...
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1answer
22 views

Is there an explicit formula for $\left(xx^T\right)^{-1}$ with $x\in\mathbb{R}^n\setminus\left\{0\right\}$?

Let $x\in\mathbb{R}^n\setminus\left\{0\right\}$. Obviously, $$A:=xx^T$$ is symmetric and positive definite. Hence, $A$ is invertible. Can we find an explicit formula for $A^{-1}$?
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59 views

Proof that for all symmetric matrices $A$ and $B$, $AB=(BA)^T$.

Recall that a matrix, $M$, is said to be symmetric if and only if $M=M^T$. I've been trying to use the homomorphic nature of the transpose operator to prove this proposition but this approach hasn't ...
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1answer
39 views

How to express double orthogonal complement?

Let $V$ be a Hilbert space and $U \subseteq V$. Then $U^\perp = \{\mathbf{v} \in V|\forall \mathbf{u} \in U, \langle \mathbf{u}, \mathbf{v} \rangle = 0 \}$. My question is, how do you express ...
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2answers
49 views

Linear algebra: What is the difference between homogenous and particular solutions?

First, I would like to mention I'm new to asking questions here, though I have found many answers here! I hope to get more involved here over time, I really like this site. If you have any suggestions ...
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1answer
22 views

Prob. 3, Sec. 4.2 in Erwin Kreyszig's Functional Analysis: How to show that $\lim\sup$ is sublinear?

Let's consider the real space $\ell^\infty$ of all bounded sequences of real numbers. Let $p \colon \ell^\infty \to \mathbb{R}$ be defined by $$p(x) \colon= \lim\sup_{n \to \infty} \xi_n \ \mbox{ for ...
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2answers
26 views

How to compute the projection of a polyhedron

Suppose that we have a polyhedron in $(x,y)$: $P=\{ (x,y) \mid A_1 x +A_2 y \leq b \}$ How can I find the polyhedron $P_x=\{ x \mid (x,y)\in P \}$? In other words, I would like to write $P_x=\{x ...
2
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1answer
28 views

The deconposition of $\mathfrak{so}(V \oplus V^*)$

Let $V$ be an n dimensional real vector space and $V^*$ be the dual vector space. We have a non degenerate inner product $(\centerdot,\centerdot)$ in $V\oplus V^*$ such that $(v+\xi , ...
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1answer
21 views

How do I extrac the anisotropic part of a tensor?

Given the elements $\chi_{ij}$ of a tensor in cartesian coordinates, with \begin{pmatrix} \chi_\bot& 0 &0 \\ 0 & \chi_\| &0\\ 0&0 & \chi_\| \end{pmatrix}, where the ...
0
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1answer
22 views

Tensor product of 2 vectors

Let be V,W 2 K-vector spaces and $a \in V, b \in W$.a,b are vectors. We know that $a\otimes b=0 \in V \otimes W$. Proof that a=0 or b=0. From definition it will be a matrix with elements $a_i b_j$ and ...
4
votes
2answers
73 views

Proof of determinant formula

I have just started to learn how to construct proofs. That is, I am not really good at it (yet). In this thread I will work through a problem from my Linear Algebra textbook. First i will give you my ...
5
votes
1answer
45 views

Group action on a category

Motivating example: We get a functor from the category of real vector spaces to the category of complex vector spaces by complexifying (i.e. tensoring over $\mathbb{R}$ with $\mathbb{C}$). Let ...
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1answer
39 views

Proof for the existence of basic feasible solution

I am trying to understand a proof for If F is non empty, the it has a BFS, where F = {x belongs to R: Ax=b, x>=0}, The proof goes likes this, first we collect all the indices(j) where xj > 0, then ...
2
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0answers
20 views

If $H$ is positive definite and $s^Ty>0$, then $s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1$

Let $H\in\mathbb{R}^{n\times n}$ be symmetric and positive definite $s,y\in\mathbb{R}^n$ with $s^Ty>0$ How can we show, that $$s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1\;?\tag{1}$$ ...
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2answers
28 views

Diagonally dominant matrix — geometric interpretation

I like to have a visual interpretation of mathematical concepts. This is simple for many important kinds of matrices: orthogonal matrices are rotations, diagonal matrices scale along the natural ...
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2answers
94 views

What is that matrix?

Let an inner product on $\mathbb{R}^n$ be given by its Gramian matrix $G$. Let $A:\mathbb{R}^n \rightarrow \mathbb{R}^k$ be a linear operator with $\mathop{\rm rank} A=k$ (We denote its matrix also by ...
2
votes
1answer
43 views

Which of the following are subspaces

I was working on subspace and found a problem that check following are subspaces of $\Bbb{R}^3(\Bbb{R})$ or not. 1) W = {($a^5$,0,0) : a∈$\Bbb{R}$} 2) U = {($a^2$,0,0) : a∈$\Bbb{R}$} My Attempt : I ...
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1answer
26 views

Linear transformation, change of basis

Linear operator $A$ in standard basis has matrix: $\left(\begin{array}{ccc} 2& 5& −3\\ −1& 2& 1\\ 2& −3& 2 \end{array}\right)$, Find its matrix in base $\cal F = ($$f1, f2, ...
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2answers
37 views

Does a decrease of an entry of symmetric non-negative matrix decrease the norm?

I think the assertion would be false but I do not see an easy example. Or is it true obviously? Suppose we have a symmetric $A=[a_{ij}]$ with all entries non-negative. Now if we decrease a $a_{ij}$ ...
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3answers
77 views

Find the necessary and sufficient condition for $A^m\to0$

Let $A$ be $n\times n$ matrix on $\mathbb{C}$. Find a necessary and sufficient condition for $A^m\to0$ as $m\to\infty$. My thought: I think it should be that eigenvalues of $A$ are less than $1$. ...
4
votes
8answers
138 views

Is $y=mx+b$ linear?

Consider $f(x) = mx+b$. Let $b\ne 0$ If $f$ is linear, $f(0)$ should yield $0$ $f(0) = m(0)+b = b$ Therefore $f(x)=mx+b$ is nonlinear. Question: Why is $y=mx+b$ called a "linear equation"? ...
3
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4answers
41 views

Determine if a point is within a section of an octagon

I've been looking at other answers on this Exchange, such as this one. My math is fairly average, but working in Cartesian planes seems so long ago... My question is this: how to determine which ...
0
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1answer
43 views

Decide the range of eigenvalues for $A+B$

Let $A,B$ be $n\times n$ Hermitian matrices on $\mathbb{C}$ such that all eigenvalues of $A$ lie in $[a,a']$ and all eigenvalues of $B$ lie in $[b,b']$. Show that all eigenvalues of $A+B$ lie in ...
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1answer
23 views

Linear transformation, image [on hold]

Linear space $\rm L$ is made from polynomials with real coefficients, whith maximum degree $2$. Is given basis $\rm e$ for space $\rm L: e_1 = 1, e_2 = x, e_3 = x^2$, also is given an image $\rm A$ in ...
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3answers
42 views

How come least square can have many solutions?

I know there always exists a least-square solution $\hat{x}$, regardless of the properties of the matrix $A$. However, I keep finding online that least-square can have infinitely many solutions, if ...
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0answers
18 views

What is meant by “homogenous problem” exactly?

Let us look at an entirely linear problem with operator $L$. For an algebraic equation $Lu=0$ is a homogenous equation. If $L$ is a differential operator (PDE or ODE) it has to be supplemented with ...
4
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1answer
616 views

Inverse of Symmetric Matrix Plus Diagonal Matrix if Square Matrix's Inverse Is Known

Let $A$ be an $n \times n$ symmetric matrix of rank $n$ with known inverse $A^{-1}$. Let $D$ be a diagonal matrix with the same dimensions and rank. What is the fastest way to compute $(A+D)^{-1}$? ...
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2answers
17 views

Linear transformation, base change

Linear operator A in standard basis has matrix: 2 5 −3 −1 2 1 2 −3 2 Find its matrix in base f1 = (1, 1, 1), f2 = ...
3
votes
2answers
100 views

Bound the norm of the partial trace of an operator on a Hilbert space

Let $H=H_1 \otimes H_2$ a composite Hilbert space and let $A, B$ bounded linear operators on $H$, and we can assume they are trace class. Let $A_2$ we denote the operator on $H_2$ obtained by taking ...
3
votes
1answer
103 views

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$

Compute $(-1)^n\sum_{k=1}^n (-1)^k\frac{(k+n-1)!}{(k-1)!(k-1)!(n-k)!}$ Define $a_{k,m}=\frac{(-1)^{k+m}(n+k-1)!(n+m-1)!}{(k+m-1)[(k-1)!(m-1)!]^2(n-m)!(n-k)!}$ Compute ...
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0answers
49 views

Advanced Linear Algebra [on hold]

Let F be any field. Let $f_1$, $f_2$, $f_3$ be the following three polynomials in $F[X]=P(F)$ $$f_1=X+1$$ $$f_2=X^2-1$$ $$f_3=X^2+3X+1 $$ Do $f_1$, $f_2$ and $f_3$ form an $F$-basis for $P_1(F)$, ...