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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Prove that if $v$ is an eigenvector, then $A^2v=c^2v$

Prove that if $v$ is an eigenvector for the matrix $A$, then $A^2v=c^2v$ Pretty much all I have is: $Av=cv$ where $v$ is a nonzero vector
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56 views

Use DeMoivre's formula to derive the trig identities

Use DeMoivre's formula to derive the trig identities $\cos3\theta = (\cos\theta)^2-3\cos\theta(\sin\theta)^2$ $\sin3\theta=3(\cos\theta)^2\sin\theta-(\sin\theta)^3$ and note: $(a+b)^3 = a^3+3a^2b+...
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54 views

if T and S are two normal operators on real inner product space also T ans S commute .Is TS normal?

I know result is true for complex inner product space because we can diagonalize Tand S. but in real inner product space T and S cannot be diagonalized.then is this result true ?how I proceed I don ...
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56 views

Adjoint operator and basis

Let ${e_1,e_2}$ be the standard basis in $\mathbb R^2$. Suppose $T$ is a linear mapping on $\mathbb R^2$ such that $T(e_1) = −2e_2$ and $T(e_2) = e_1 − 3e_2$. Find a basis in $\mathbb R^2$ in which ...
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1answer
50 views

Linear independence of Eisenstein Space Basis

I'm currently working throught Diamond and Shurman's book "A First Course on Modular Forms". On page 129 Theorem 4.5.2 states Let $ N $ be a positive integer and let $ k \geq 3 $. The set $$ \...
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2k views

Does an overdetermined system always have no solutions? [closed]

What is the problem with over-determined systems in linear algebra? Do they always have no solution? Is there a proof of that?
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539 views

Need help finding a specific transition matrix with polynomials.

I am not told it is a transition matrix till later and the book never describes a transition matrix, it just tells me to find $A$ from the equation $[w]_{B}=A[w]_{C}$ where $w$ is an arbitrary matrix ...
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18 views

How to find the all the solutions of the linear iterated integer equation

I have following problem which needs some special tricks. Let $a_1=2, b_1=b_2=1$ define $$ a_n=2a_{n-1}+b_{n-1}+2c_{n-1}$$ $$ b_n=a_{n-1}+b_{n-1}+c_{n-1}$$ $$ c_n=a_{n-1}+b_{n-1}+c_{n-1}$$ then how ...
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60 views

Eigenvalues of a matrix whose square equals its transpose

Let $A$ be a $n\times n$ matrix with $A^2=A^t$. Show that every real eigen value of $A$ is either $ 0$ or $1$
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43 views

unnecessary constraint in optimization problem

I have some optimization problem (optimizing parameter $\alpha$)with those constraints: $$\alpha_i\ge0$$ $$\sum\limits_i \alpha_i y_i =0$$ and a third constraints: $$w-\sum\limits_i \alpha_i y_i x_i = ...
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20 views

Show that $(2proj_v-I)^T(2proj_v-I)=I$

So we have that $(2proj_v-I)^T(2proj_v-I)=I$ Working on the left-hand side we have: $$(2proj_v^T-I^T)(2proj_v-I)$$ Multiplying through we have: $$(2proj_v^T)(2proj_v)-2Iproj_v+I$$ But I don't ...
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3answers
44 views

The subspace $S ⊆ \mathbb{R}^n$ has linearly independent vectors $u_1,…u_k$. Show that any basis for $S$ must have at least $k$ vectors.

Let $S ⊆ \mathbb{R}^n$ be a subspace. Say that $u_1,.....u_k ∈ S$ are linearly independent vectors. Show that any basis for S must have at least k vectors. "Say that $u_1,.....u_k ∈ S$ are linearly ...
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1answer
62 views

$AM=I$, where $M$ is a rectangular matrix with full column rank, prove that $A=M^+$?

$AM=I$, where $M$ is a rectangular matrix with full column rank, then $A=M^+ $(Moore-Penrose pseudoinverse)?
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1answer
40 views

Axler's proof of existence for the polar decomposition

I'm reading Axler's proof of the existence of the polar decomposition of a linear operator, and he starts by positing the existence of a linear map, $S_1$, such that $S_1(\sqrt{T^*T}v) = Tv$ (see ...
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1answer
125 views

left eigenvector and similarity matrix of Jordan canonical form

Let $\mathbf{L}$ be a $N\times N$ matrix and $\mathbf{L}\mathbf{P}=\mathbf{P}\mathbf{J}$ where $\mathbf{J}=[j_{ik}]$ is Jordan block matrix. If $~j_{NN}=0$ is a "simple'' eigenvalue of $~\mathbf{L}$...
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34 views

Matrix Transformation across multiple planes

Let $T_1$ be a reflection of $\Bbb{R}^3$ in the xy plane, $T_2$ is a reflection of $\Bbb{R}^3$ in the xz plane. What is the standard matrix of transformation $T_2T_1$? Here's my thinking so far: ...
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1answer
351 views

Adjoint Operator and Inverse

I am solving the following question and I am not really sure about the way I approach Question 1: Assume that $T:U\rightarrow U$ is invertible map. Prove that $(T^*)^{-1}=(T^{-1})^*$ Here is my ...
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1answer
56 views

Linear Algebra reflection question

Let $T_1$ be a reflection of $\mathbb{R}^3$ about the plane $x = y$ and $T_2$ be a reflection of $\mathbb{R}^3$ about the plane $x = z$. Find standard matrix for the transformation $T_2 \cdot T_1$.
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Show that an orthogonal transformation from $\mathbb{R^3}$ to $\mathbb{R^3}$ can be written as the composition of reflections

If $T$ is an orthogonal transformation from $\mathbb{R^3}$ to $\mathbb{R^3}$, then $T$ may be written as the composition of three or fewer reflections about planes in $\mathbb{R^3}$. Suppose $\{x_1,...
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Given two non-negative vectors $r,c$, is there always a non-negative matrix A whose marginals are $r$ and $c$?

Let $A$ be an nxm matrix. We can easily determine its row and column marginals $r$ and $c$: $r=A1$ $c=1^TA$. Suppose however, that you are given non-negative marginals $r,c$. Is there always a ...
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The reflection onto a hyperplane $V\in \mathbb{R^n}$ is orthogonal

We have that $ref_V(\vec{x})=2proj_V\vec{x}-\vec{x}$. So that $ref_V(\vec{x})=\vec{x}+2(proj_V\vec{x}-\vec{x}) = 2proj_V{\vec{x}}-\vec{x}=(2proj_V\vec{x}-I)\vec{x}$ It's supposed to be that this ...
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1answer
76 views

Show that mutually orthogonal subspaces of $\mathbb{R^n}$ must have a collective dimension $\le n$

Suppose $V_1,...,V_k$ are mutually orthogonal subspaces of $\mathbb{R^n}$ for $k\ge 2$. Show that $\dim(V_1)+...+\dim(V_k)\le n$. I was trying to take the direction of using induction, because I'...
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60 views

If W is a subspace of V then $W^⊥$ is a subspace of $V^⊥$?

I believe this to be false. As it seems that $V^⊥$ is a subspace of $W^⊥$ instead, but can someone explain more in depth?
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1answer
75 views

Finding the Moment Generating Function given f(x)

I think I'm having trouble with this because of the absolute value. Otherwise, I know how to solve for the moment generating function. The problem is that show that if a random variable has the ...
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1answer
23 views

Condition for $n+1$ vectors to be in general position?

We say that points/vectors $v_0,\ldots v_n$ are in general position if the set $\{v_1-v_0,\ldots, v_n-v_0\}$ is linearly independent. If it is the case that any $n$ of our $n+1$ vectors are ...
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1answer
29 views

Transformations that are not orthogonal

Are there transformations that will preserve angles but are not orthogonal? Intuitively it seems like this should be true (i.e. we can have a transformation preserve angles without being orthogonal), ...
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35 views

Show that $(A\vec{x})\cdot (A\vec{y})=\vec{x}\cdot \vec{y}\rightarrow \|A\vec{x}\|=\|\vec{x}\|$

Since we know the first expression is equivalent to $A^TA=I_n$, I was thinking we can multiple by $\vec{x}$ on both sides to get $A^TA\vec{x}=\vec{x}$ and once we take the magnitude on both sides we ...
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1answer
22 views

Show that $A^TA=I_n\rightarrow (A\vec{x})\cdot (A\vec{y})=\vec{x}\cdot \vec{y}$ $\forall \vec{x},\vec{y} \in \mathbb{R^n}$

I start by assuming that $A^TA=I_n$ holds, but I don't know how to obtain the right-hand side from this. Some ideas I've had were multiplying both sides by $\vec{x}$, but I still don't see how this ...
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1answer
25 views

set of solutions of inhomogenous linear equation

I want to solve the following exercise: Given is a linear map $F: V \rightarrow W$. I want to show that if F(x) = b has some solution $x_0$, then the set of solutions of the inhomogeneous equation is ...
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97 views

True or False The image of the unit square under one-to-one matrix operator is a square [closed]

The image of the unit square under one-to-one matrix operator is a square is this true or false
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187 views

Prove that x is in W if and only if $proj_W$(x) = x

Let W be a subspace of $R^n$, and let x be a vector in a $R^n$. I need to prove that x is in W if and only if $proj_W$(x) = x Can I have some hints on how to get started with this problem?
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25 views

Eigenvectors of the square of the matrix

Given $A: n\times n$ matrix with eigenvector $w$ for eigenvalue $c$, does $B$, where $B^2 = A$ have $w$ as an eigenvector? I.e, $A*w = B*B*w = c*w$. Is $w$ an eigenvector of $b$ with eigenvalue $\...
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1answer
122 views

Conjecture that $A^{T}BA = ABA^{T}$ for any symmetric matrix $B$ in $\mathbb{R}^n$

While trying to understand the Kalman filter, and by experimentation with Python I came up with the conjecture in the title. First of all is it true? Second, if it is, how can I prove this? I would ...
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3answers
94 views

Solve for $\textbf{x}(t)$ from the system of differential equations $\textbf{x}' (t) = A \textbf{x}(t)$.

Solve for $\textbf{x}(t)$ from the system of differential equations $\textbf{x}' (t) = A \textbf{x}(t)$, where $$ A =\begin{bmatrix} 0 & -1 & 0\\1 & 0 & 0\\ 0 & 0 & 1\\\end{...
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1answer
25 views

How does minimum squared error relate to a linear system?

Given some system $U*x = b$, I've solved for $x^*$, the least squares solution. I then compute the minimum squared error by $||U*x^* - b||^2$. I know that the least squares solution minimizes the ...
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0answers
109 views

How to derive the following from Azuma's inequality?

This is claimed in Proposition 1 in the paper http://arxiv.org/abs/1409.6110 Let $A$ be a $n \times d$ matrix. $A$ can have only $K$ different types of rows i.e. rows of $A$ are chosen from a set of ...
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1answer
24 views

Infinite homogeneous system and linear independence

For $J \geq K$, consider a $J \times K$ matrix $M$ created by stacking row vectors $M(j)$ for $j=1,...,J$. We know the following are equivalent: (a) The homogeneous system $M z=0$, where $z$ is a $K$-...
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651 views

Can basis vectors have fractions?

So I was diagonalizing a matrix in a book, and one of the basis vectors was [3/2, 1], after doing the problem, the answer in the book was different than mine. It came with an explanation, and in it ...
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57 views

If A is some invertible $n \times n$ matrix then show $\det(A^n) = (\det(A))^n$ for all $n\in \mathbb{Z}$

So there exists $A^{-1}$. I am assuming $\det(AB)=\det(A)\cdot\det(B)$ and $(A^d)^f=(A^{df})$ I know the proof for $\det(A^{-1})=(\det(A))^{-1}$ is: $\det(I_n)=1$ $\det(A\cdot A^{-1})=1$ $\det(A)\...
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1answer
150 views

minors and rank of a matrix

When reading a text, I came across a statement saying "the rank of an $m\times n$ matrix is $r$ if and only if all $(n-r+1)\times(n-r+1)$. minors vanish" Could anyone explain what it means by a ...
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28 views

Modules over PIDs

Let $M=\mathbb{Z}^4/N$ where $N$ is a subgroup of $\mathbb{Z}^4$ generated by $(1,0,-1,3)$ and $(2,4,8,-6)$. Recognize $M$ as a product of cyclic groups. Here I have to use the following Theorem: If ...
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1answer
64 views

Permutation Matrices for n = 5

Let $\sigma \in S_n$ denote the permutation given by $$ \sigma = \left( \begin{matrix} 1 & 2 & 3 & \ldots & n \\ n & 1 & 2 & \ldots & n-1 \end{matrix} \right) $$ and ...
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135 views

How does the representation of co-vectors change if we change the basis of a vector space $V$?

I'm trying to understand how vectors, differential forms and multi-linear maps in general transform under change of coordinates. So I start with the simplest case of vectors. Here's my own attempt, ...
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the exact graph of the general solution for $x'=\begin{bmatrix} 1 & 1\\ 4& 1 \end{bmatrix}x$

i need someone to give me exact graph of the general solution for $$x'=\begin{bmatrix} 1 & 1\\ 4& 1 \end{bmatrix}x$$ i solved it manually , the general solution is like this $$x(t)=c_1\...
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1answer
35 views

Linear Transformation: When T(v)=v

I have a homework to do in which with a pre-defined transform I have to find a vector v that after the transformation equals itself: $T(v)=v$. The transformation happens from $\mathbb{R}^3$ to $\...
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2answers
86 views

Determinant matrix proof

Let $A$ be an $n\times n$ matrix and $i,j,k$ be $1\leq i,j,k\leq n$ and $\alpha,\beta \in \mathbb{R}$. I am supposing that $\bf{a}_k$(the $k$-th row) is equal to $\alpha \bf{a}_i+\beta \bf{a}_j$. ($\...
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1answer
25 views

$H=[h_1, h_2, \cdots, h_n]$, $h_i\in \mathbb{C}^m, m>n$. prove the orthogonal complement problem

$H=[h_1,h_2,\cdots,h_n]$, where $h_i\in \mathbb{C}^m, m>n$. Let $Q_i$ be the matrix whose columns are formed by the orthonormal bases of the orthogonal complement of the subspace spaned by $\{h_1,...
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1answer
73 views

abelian group as Z module

How Would you prove that every abelian group can be understood as a Z-Module in a unique way? I would guess that you would have to prove its bijective, but not sure how to go about this
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133 views

Orthonormal basis Parsevals identity.

Let $O={u_1,...,u_k}$ be an orthonormal set in $V$. Prove that $O$ is an orthonormal basis if and only if Parseval's identity holds for all $v,w \in V$ i.e if and only if $$\langle v,w\rangle=\sum_{...
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1answer
35 views

Symmetric matrices with the eigenvalues comparable

Let $A,B$ be $n\times n$ real symmetric matrices, with eigenvalues $\lambda_i$ and $\mu_i$ respectively, $i=1,\cdots,n$. Suppose that $$\lambda_i\leq\mu_i,\forall\ i.$$ Show that there exists an ...