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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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
21 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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2answers
18 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
12 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 $U$ ...
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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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1answer
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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1answer
30 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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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 ...
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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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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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
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 ...
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0answers
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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3answers
72 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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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
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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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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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
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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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 ...
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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 ...
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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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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 ...
5
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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 ...
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 $$s^THs-\frac{s^Tyy^Ts}{s^Ty+y^TH^{-1}y}\ne -1\;?\tag{1}$$ ...
5
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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 ...
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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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 ...
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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 ...
4
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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"? ...
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3answers
76 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 $1$. ...
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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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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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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 ...
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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 ...
2
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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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2answers
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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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 ...
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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 = ...
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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 $P_1(F)$, ...
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3answers
34 views

How to determine a basis and the dimension for this vectorspace?

Determine a basis and the dimension for the following vectorspace: \begin{align*} W = \left\{A \in \mathbb{R}^{3 \times 3} \mid A \ \text{is a diagonal matrix and} \ \sum_{i=1}^3 A_{ii} = 0\right\} ...
2
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1answer
21 views

different ways to see why this matrix limit is correct

given that $0 < a < 1$ it follows that: $$\lim_{n\to\infty}\begin{pmatrix} a & (1-a) \\ (1-a) & a \end{pmatrix}^n = \begin{pmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 ...
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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 ...
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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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0answers
17 views

Formula Index Confusion

I am working on a computer vision project and need to implement the formula on the bottom of page 13 of http://www.dgp.toronto.edu/~donovan/stabilization/opticalflow.pdf My question pertains to the ...
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1answer
14 views

If complex matrix 2*2 has a real eigenvalue then matrix of its conjugate elements has a real eigenvalue too

If $\begin{pmatrix}a&b\\c&d\end{pmatrix}$ $\in$ $\mathbb C^{2x2}$ has a real eigenvalue then $\begin{pmatrix} \overline a& \overline b\\ \overline c&\overline d\end{pmatrix}$ $\in$ ...
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0answers
18 views

Linear operator, image, transformation

Linear space L is made from polynomials with real coefficients, whith maximum degree 2. Is given basis e for space L: e1 = 1, e2 = x, e3 = x^2, also is given an image A in this space A(P(x)) = P(x + ...
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1answer
15 views

Question on relation between convex sets

The book I am referring regarding bell inequalites has the following two things it mentions about two convex sets which I am unable to understand. There are two convex sets $C$ ($C$ is a polytope ) ...
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1answer
37 views

Checking if a matrix is in the span of other matrices

Problem: Expand the following set matrices \begin{align*} \left\{ \begin{pmatrix} 1 & -1 \\ 2 & 3 \end{pmatrix}, \begin{pmatrix} 2 & 1 \\ -1 & 4 \end{pmatrix}, \begin{pmatrix} 0 & ...
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0answers
14 views

Normal Vectors to Action of Orthogonal Group

Let $X\in\mathbb{R}^{n\times r}$ be a fixed matrix with orthogonal columns, and let $U\in\mathbb{R}^{n\times r}$ be given. Because the group of orthogonal $r\times r$ matrices, $O(r)$, is a compact ...