Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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Geometry Transformations (Linear)

If someone could run quickly through the theory and methods on this it would be hugely appreciated. Thank you. - ...
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Is it possible to find

If $\frac {(a-b)(c-a)}{(b-c)(d-c)}$=$\frac {2012}{2013}$ then find the value of $\frac {(a-c)(b-d)}{(a-b)(c-d)}$ in terms of numbers Note: a,b,c,d are real numbers
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Introduction to Linear Algebra 4th Edition by Gilbert Strang fully written solutions / or another book with fully written solutions!

I have gotten my hands on the following book Introduction to Linear Algebra 4th Edition by Gilbert Strang and it's not sufficient for my learning needs, at least not on it's own. I have access to the ...
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Two affine subspaces parallel

Theorem: Two affine subspaces $V,V'$ of $(X,\overrightarrow{X})$ are said to be parallel $(V\parallel V')$ if there is a translation such that $t_{\vec{u}}(V)=V'$ $V\parallel V' ...
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1answer
26 views

Convex decomposition of a vector

Let $(a_i)_{i=1}^n$ be a probability vector, that is, $a_i\geq 0$ and $\sum_i a_i=1$ and let $(U_{ij})_{i,j=1}^n$ be a unistochastic matrix, that is, the pointwise square of a unitary matrix. Now ...
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2answers
53 views

New proof about normal matrix is diagonalizable.

I try to prove normal matrix is diagonalizable. I found that $A^*A$ is hermitian matrix. I know that hermitian matrix is diagonalizable. I can not go more. I want to prove statement use only this ...
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2answers
32 views

Visulizing column/row space and null/left null space, A and x

So here is a direct passage from the book we are learning from. Now the image I get in my mind is that A belongs to column space, A(transpose) belongs to row space and that x belongs to null ...
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1answer
23 views

Resolvent matrix

Suppose $A$ is a triangular matrix. What is the most efficient known algorithm to compute the polynomial (in $x$) matrix $(xI-A)^{-1}$? Of course, $(xI-A)^{-1}= N(x)/p_A(x)$, where $p_A$ is the ...
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42 views

Find the eigenvalues of the following matrix

Consider $A =\left( \begin{array}{ccc} -1 & 2 & 2\\ 2 & 2 & -1\\ 2 & -1 & 2\\ \end{array} \right)$. Find the eigenvalues of $A$. So I know the characteristic polynomial is: ...
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1answer
18 views

Given multiple polynomial equations find a basis.

I have read several other threads on Math.SE, including the similarly titled: basis of the polynomial vector space I've also checked out a video lecture on Youtube by njwildberger, but I simply have ...
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A question on vectors represented by multilinear polynomials

Let the set of multilinear polynomials of degree atmost $t$ in $\Bbb Z[x_1,\dots,x_n]$ be $\Bbb Z^{t}[x_1,\dots,x_n]$. Let $S=\{0,1\}^n$. Fix an ordering of $S$. For every $f\in\Bbb ...
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Uniqueness of k-sparse solution in compressed sensing

This question has been bothering me for some time now. Please correct me if I've got anything wrong. In compressed sensing, we try to find a unique k-sparse solution to an underdetermined system by ...
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3answers
36 views

Prove $A_1, A_2, \ldots,A_{\ell-1}, A_{\ell+1}\ldots \ldots, A_{m} $ in $\mathbb R^m$ are linearly independent viewed as vectors in $\mathbb R^{m-1}$?

Suppose I have linearly independent vectors $A_1, A_2, \ldots, A_m$ in $\mathbb R^m$ Consider the matrix $B = [A_1, A_2, \ldots A_m]$ consisting of these vectors and suppose $B^{-1}[A_1, A_2, ...
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1answer
25 views

“a matrix is positive semi-definite” not necessarily equavalent to “all leading principle minors are nonegative”?

Have a look at this matrix: $$ H = \left( {\begin{array}{*{20}{c}} 0&0&1&0\\ 0&0&0&1\\ 1&0&0&0\\ 0&1&0&0 \end{array}} \right).$$ All the leading ...
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29 views

Lyapunov function

How to do this problem? Find a Lyapunov function for $(0,0)$ in the system: $$x˙=3xy^2−11x^2$$ $$y˙=11x^3−4y^3$$ I know there is no formula for finding Lyapunov functions for a system, so how do I ...
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1answer
24 views

Linear algebra eigenvalue proof

I asked this question before but after thinking about it I was trying to think of another way. Proof multiplied complex matrix has non negative eigenvalues I need to show given that A is a matrix $\in ...
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3answers
39 views

Formulation of the matrix determinant

How was the idea (and the equation) for the determinant of a square matrix formulated, and why does it work? All I've learned is that the determinant of a matrix is 0 when some row is a linear ...
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1answer
17 views

Proof multiplied complex matrix has non negative eigenvalues

I need help to show given that $\mathbf A$ is a complex nxn matrix that $\mathbf {AA}$* is a Hermitian matrix and the eigenvalues > 0 Where * is taking the transpose of the complex conjugate of the ...
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1answer
41 views

Prune a linearly independent set? What is the element of Span(Z)? What is the

Consider the following subset of $P_3(\mathbb{R})$ (real polynomial functions of degree at most 3). $$ Z = \{f_1, f_2, f_3, f_4, f_5\} $$ where $f_1(x) = 1-2x+2x^2-x3$, $f_2(x) = 1-x+x^2+x^3$, ...
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1answer
30 views

Solution space for a set??

Make a system with 3 eqns and 3 variables of which the solution space is spanned by V =[ 1 3 0], [1 0 -1]? Do I do the cross product of these two? Can someone show me how to do this? Also Can you ...
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dimension of vector space $\frac{\langle e_{ab_1\ldots b_p}\rangle}{\langle \sum_{1\leq i\leq p}e_{ab_1\ldots \widehat{b_i}\ldots b_pc}\rangle}$

Let $p$ be a prime and $n\!\in\!\mathbb{N}$. What is the dimension of the $\mathbb{Z}_p$-module $$V_{p,n}=\frac{\langle e_{ab_1\ldots b_p};\: 1\leq a<b_1<\ldots<b_p\leq n\rangle}{\langle ...
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1answer
17 views

This is only a subspace if $b=0$ - Axler - LADR p13

I have written here in Axler - Linear Algebra Done Right, page $13$. If $b\in \mathbb{F}$, then $\{(x_1,x_2,x_3,x_4)\in \mathbb{F}^4: x_3 = 5x_4 + b\}$ is a subspace of $\mathbb{F}^4$ if and only ...
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Find the determinants of the given matrices

Consider scalars $a,b,c,d,e,f$ such that $\det\left( \begin{array}{ccc} a & 1 & d\\ b & 1 & e\\ c & 1 & f\\ \end{array} \right) = 7$ and $\det\left( \begin{array}{ccc} a & ...
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Lovasz number and bipartite complement

Let $G=(V,E)$ be a graph on $n$ vertices. An ordered set of n unit vectors $U=(ui|i∈V)⊂R^N$ is called an orthonormal representation of G in $R^N$, if $u_i$ and $u_j$ are orthogonal whenever vertices i ...
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1answer
17 views

Proving Equality of Hermitian matrix

I need to show that the equality $\ A \mathbf u \cdot \mathbf v = \mathbf u \cdot A\mathbf v \ $ holds true for all $\mathbf u, \mathbf v \in \mathbb{C^n}$ if and only if the matrix A is Hermitian ...
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5answers
290 views

Is the determinant of this matrix positive or negative?

$\left( \begin{array}{ccc} 1 & 1000 & 2 & 3 &4\\ 5 & 6 &7&1000 &8\\ 1000&9&8&7&6\\ 5 & 4&3&2&1000\\ 1&2&1000&3&4\\ ...
2
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1answer
29 views

Every skew-symmetric matrix has a non-negative determinant

I'm breaking this up into the even case and odd case (if $A$ is an $n\times n$ skew-symmetric matrix). So when $n$ is odd, we have: $\det(A)=\det(A^T)=\det(-A)=(-1)^n\det(A)\Rightarrow ...
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1answer
18 views

Basis and dimension of the subspace of solutions to $A\mathbf{x}=\mathbf{0}$

Consider $$ A =\left( \begin{matrix} 1 & -1 & 0 & -2 \\ 0 & 0 & 1 & -1 \\ \end{matrix} \right) $$ and find a basis and the dimension of $S(A,0)$, where $S(A,0)$ is the ...
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Link between the projection onto a subspace and projection onto hyperplane

The projection onto a hyperplane $H=\{x\in \mathbb{R}^n|\langle a,x\rangle=b\}$ is defined to be $P_{H}(x)=x-\frac{\langle a,x\rangle-b}{||a||^2}a,$ and characterized by $\langle c-p,x-p\rangle\leq0.$ ...
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Double circle representation using ideal rubber bands

Let $V$ be a 3-connected planar graph, $V^*$ be its dual graph. Let $(C_i : i \in V)$ and $(D_p : p \in V^*)$ be a collection of circles so that if any edge $\{i,j\}$ borders any faces $a$ and $b$, ...
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2answers
34 views

Proof that every subspace is a vector space

I was unable to find a simple proof that a subspace is a vector space. I know that a subspace $S$ is a subset of a vector space, such that: $$\vec 0 \in S\\\vec a + \vec b \in S\\\alpha\vec a \in S$$ ...
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1answer
53 views

Equation. Proof that there is no solution.

Prove that $\nexists x,y \in \mathbb{N} $ such that: $$\begin{cases} x -y - 3k -1 = 0 \\ x-y -4l -2 =0 \\ x+y - 3f - 2 =0\\ x + y - 5m - 2 = 0 \\ x,y \in \mathbb{N} \end{cases} $$ I'm asking form any ...
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Prove that $L(V,W)$ is a vector space over $K$

Let $V$ and $W$ are $k$-vector spaces and let $L(V,W)$ be the set of linear maps $V\to W$. I have to prove that $L(V,W)$ is a vector space over $K$. I have already done a step by step proof ...
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1answer
30 views

Is the intersection between two $n$-spheres an $(n-1)$-sphere?

It is true that the intersection between two $n$-sphere in $\mathbb{R}^n$ is a $(n-1)$-sphere if is not empty or a single point? I have tried to prove it but my only idea is to work with equations and ...
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1answer
32 views

Prove a certain matrix is positive semidefinte.

Consider a stochastic matrix $P$, i.e. real, non-negative, square, rows sum to one. Consider $\Xi$ to be a diagonal matrix with a principal left eigenvector of $P$ on the main diagonal and zeros ...
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1answer
20 views

Show matrix is element in eigenspace

Let $A$ be an $n\times n$ matrix such that $A^2=A$. a) Let $E_{1}(A)=\{x \in \mathbb{R^n} | Ax=x \}$: let $E_{0}(A)=\{{ x \in \mathbb{R^n} | Ax=0\}}$. Let $x$ be any vector in $\mathbb{R^n}$. Show ...
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4answers
122 views

Rank of a matrix $A^2$ without calculating the square

I have a matrix $A=\begin{bmatrix} 2 & 0 & 4\\ 1 & -1 & 3\\ 2 & 1 & 3 \end{bmatrix} $ with rank 2. How do I prove that the matrix $A^2$ has also rank 2 without actually ...
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0answers
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Find coordinates of point that satisfy given conditions

I have A(1,2,3) , B(-1,0,1), C(1,-1,1) which are points in $\mathbb{R}^3$. I'm trying to find another point H such that AH${\parallel}$AC and BH${\perp}$AC. I set H = ($h_1$, $h_2$, $h_3$), and took ...
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1answer
19 views

Spectral decomposition - generalized eigenspaces

Suppose we have a real $n\times n$ matrix $A$ with spectrum $\sigma(A)=\{\lambda_1,\lambda_2\}$ (with $\lambda_1, \lambda_2$ discrete). Also, we have $alg\,mult\, \lambda_i\neq geo\, mult \,\lambda_i ...
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1answer
26 views

How to prove matrix geometric convergence to any matrix?

Suppose I have two vectors $x$ and $v$, and we want to calculate the following expression: $$(I+x\cdot v^{T})^{-1}$$ My professor affirmed that we could treat this as a "geometric progression" ...
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Eigendecomposition Parameterization of Real Matrix

Given a set of distinct non-real eigenvalues $\lambda_1, \dots, \lambda_N$, so that $\lambda_{2n} = \overline{\lambda_{2n+1}}$. Accordingly given a set of non-real orthonormal eigenvectors $v_1, ...
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Orthogonal basis of $R^n$

I have to prove that if orthogonal basis of $\mathbb R^n$ containing only vectors which coordinates are $1$ or $-1$ exisists then $n \leqslant 2$ or $n$ is divisible by 4. It's obvious that n have to ...
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1answer
18 views

Help solving system of linear equations.

In the process of running through an algorithm, I have derived the following systems of equations: i) $1/3 + 1/3x_1 + 1/3 x_6 = x_5$ ii) $1/2 + 1/4 x_6 = x_1$ iii) $1/2 + 1/2 x_5 = x_6$ I've tried ...
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2answers
52 views

Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$

Prove/Disprove: $v$ is an eigenvector of $T^n$ implies $v$ is an eigenvector of $T$ I'm pretty sure it's not necessarily true, but can't think of a counter example. Can you help me think of ...
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1answer
25 views

Which of the following are true?

I need to find which of the following are true? $\exists A\in M_{2\times 5}(\mathbb{R})\ni\dim$ of null space of $A$ is $2$ My ans: False as $\dim Null(A)+\dim Im(A)=5\Rightarrow\dim ...
2
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1answer
26 views

Verify that the set $\{ (b-a)^{-\frac{1}{2}}e^\frac{2\pi ixj}{b-a} \}$ is an orthonormal basis of $L^2(a,b)$.

Let $H=L^2(a,b)$ with $a<b$. Verify that the set $\{ (b-a)^{-\frac{1}{2}}e^\frac{2\pi ixj}{b-a} \}$ is an orthonormal basis of $L^2(a,b)$. Verify also that $$\{(b_1-a_1)^{-\frac{1}{2}} \cdot \dots ...
2
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1answer
35 views

understanding the matrix transpose

Can you explain me, please, what does it mean the transpose of a matrix ? I know the definition in the context of matrix theory and its generalization to adjoint operators (transpose of a linear ...
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4answers
19 views

Orthogonal nonzero vectors and linear independence. [on hold]

Show that if $\mathbf{u}$ and $\mathbf{v}$ are orthogonal nonzero vectors, then they are linearly independent.
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3answers
58 views

Question on matrix exponential

Let $A$ be a real matrix with real eigenvalues $\lambda_k$ and complex eigenvalues $\alpha_ k \pm i\omega_ k$ , all of which are simple. I'm trying to show that every element of the matrix $e^ {tA}$ ...
5
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4answers
97 views

How do I show that the derivative of the path $\det(I + tA)$ at $t = 0$ is the trace of $A$?

Here, I'm taking $A$ to be a linear operator on $\mathbb R^n$ for $n>1$. Can you please tell me how to solve such a problem?