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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Proving or disproving the existence of a vector space

Suppose the vector space $V$ is spanned by $(1,0,1,0), (0,1,0,1), (1,1,0,0)$. Is it possible to find a subspace U of $\mathbb R^4$ such that $V \subsetneq U \subsetneq \mathbb R^4$? I have checked ...
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Derivation of Simple Projectile Motion with Drag

Given the initial velocity $v_0$ and angle $\theta$ of a projectile on the ground, using Newton's second law and the acceleration due to gravity $\mathbf g=\left\langle0,-g\right\rangle$, I was able ...
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iid Gaussian random matrix $A\in M_n$ has full rank with probability 1?

I want to prove that: iid Gaussian random matrix $A\in M_n$(I mean whose elements are iid Gaussian) has full rank with probability 1 Below is my consideration: $$1-P(\text{full ...
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3answers
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Diagonalize Matrix A

I am told to diagonalize Matrix A i solved for P and P inverse ...
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25 views

How to take the derivative of a matrix with respect to itself?

Could someone please explain how to take the derivative of matrix with respect to itself? $$\frac{\partial \textbf{X}}{\partial \textbf{X}}$$ where $\textbf{X}$ is an M x N matrix
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If the Unit Vectors are equal, Are the directions equal too?

Given that the unit vector of x = unit vector of y, can we conclude that the Direction (or Sign) of x is always equal to Direction of y?
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How do I construct a Schauder basis of $l^2$ with the set $\{v_n\}

How do I construct a Schauder basis of $l^2$ with the set $\{v_n\} \subset l^2$, where $v_n=\frac{1}{\sqrt{2}}(e_n-e_{n+1})$ if $n$ is odd and $v_n=\frac{1}{\sqrt{2}}(e_n+e_{n-1})$ if $n$ even. Note ...
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How was step 1 done in Gaussian Elimination?

Suppose I have matrix $B:= \begin{bmatrix}4 & -2 & -2\\-2 & 5 & 3\\ 2 & 3 & 7 \end{bmatrix} $ Performing Gaussian Elimination we get: Step 1 ~$\begin{bmatrix}4 & -4 ...
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Is the function continuous that maps a vector to the coefficients of its expansion in a basis?

Let $V$ be a finite-dimensional vector space with basis $e_1, e_2, \ldots, e_n$. Consider the function $f:V\rightarrow\mathbb{R}^n$ defined such that, for each $v\in V$, $f(v)=(a_1,a_2,\ldots,a_n)$ ...
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an additive bijective map

Let H be a Hilbert space and $\Phi:B(H)\longrightarrow B(H)$ is an additive bijective map. If $\mathbb{R}I⊆\Phi(\mathbb{R}I)$, can we conclude by the bijectivity of $\Phi$ that ? ...
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If $\vec{v}$ is any non-zero vector perpendicular to $\vec{u}$, show that $\vec{v}$ is an eigenvector of $S$ [on hold]

I have the following problem.. I solved the first one however i can't find out how to solve the second (b). Suppose $\vec{u}$ is a unit row-vector in $\mathbb R^n$ , and $A=uu'$ matrix. (a) ...
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$x-y-2z=0$ find a perpendicular vector

Why is the vector $e=(1,-1,-2)$ ?
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Prove the automorphism given by $\phi \left(g\right)=\left(g^{-1}\right)^t$ is not an inner automorphism of $SL_n\left(R\right)$

Prove the automorphism given by $\phi \left(g\right)=\left(g^{-1}\right)^t$ is not an inner automorphism of $SL_n\left(R\right)$ Having no success with this question, I turn for your help =] I ...
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Bound spectral radius of a certain matrix.

Consider a stochastic matrix $P$, i.e. real, non-negative, square, rows sum to one. Consider $\Xi = \text{diag}(\xi)$ to be a diagonal matrix with a principal left eigenvector $\xi$ of $P$ (i.e. ...
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The dimension and basis of the set $F = \{(a+3b,a-b,2a-b,4b)| a,b \in \mathbb{R}\}$

Let $F = \{(a+3b,a-b,2a-b,4b)| a,b \in \mathbb{R}\}$ Show that F is a subspace of $\mathbb{R}^4$; Find a basis for F; Find the dimension of F. I have part A completed and showed ...
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showing something is an inner product

I'm trying to do the question above. I have found $f(v,w) = v^TAw$ where $A = \begin{pmatrix} -1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 ...
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Find sum of vector formula (?)

Find the sum of $$x(y > 10) + z(y < 30),$$ where $x$, $y$ and $z$ are vectors and $$x(y>10) = \{x_i: y_i > 10\}.$$ I have no idea where to even get started on this problem. ...
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2answers
27 views

What is the meaning of these summations?

Am I meant to add the two summations together, or multiply them? If the latter, what makes it any different from an outer product?
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Is the zero matrix diagonalizable?

Then for any invertible matrix $P$, we can say $P^{-1}\cdot 0 \cdot P=0$ ?
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Finding subspace's base

Let W be a subspace of $\mathbb{R}^4$: $ \begin{cases} x_1+2x_2+3x_3+4x_4=0 \\ 2x_1+2x_2+x_3+3x_4=0 \end{cases}$ Find base of W and extend it to the base of $\mathbb{R}^4$ How to approach this ...
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Misshap/typo in question/answer?

I am trying to understand question 3. The first image below is some information about how to solve it. Take notice of the of marked yellow area, espicially about how it becomes $1/9$. Here on ...
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1answer
22 views

Generalized Eigenvector for 4x4 matrix

I'm working on Systems of Differential Equations and I'm looking to find the Generalized eigenvector for the following matrix: $\left[\begin{array}{rrrr} 3 &-4 &1 &0 \\ 4& 3 &0 ...
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28 views

How to rewrite a derivative w.r.t. tensor as w.r.t. vector

I'm stuck on a (probably very simple) problem I've come across. Take a function $f(A)$ where $A$ is a 2-tensor. Now suppose $A=vv^T$ for an $\mathbb{R}^n$ vector, $v$. I want to rewrite the object ...
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The inverse of $(I-A)$ and the spectral radius of a nonnegative $A$ matrix

Suppost that $A$ is a nonnegative matrix, and let denote the identitiy matrix with $I$ and the spectral radius of $A$ with $\rho(A)$. Note that because $A$ is nonnegative according to the ...
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Given two linear transformations, find the preimage of a given point for the composite transformation

If someone could run quickly through the theory and methods on this it would be hugely appreciated. Thank you. Let $f: \Bbb R^2 → \Bbb R^2$ be reflection in the line $y = x$ and let $g: \Bbb ...
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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
33 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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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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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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28 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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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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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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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
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“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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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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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
42 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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21 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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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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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
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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
18 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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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\\ ...
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1answer
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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 ...