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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Linear Dependence

Hey everyone, so for this question I take the three row vectors and put them on top of each other putting $A$ on top and $B$ at the bottom. I then proceed to augment the matrix with the zero vector. ...
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62 views

What is the name of this matrix theorem?

The Theorem Consider the linear transformation $ L:R^{n}\rightarrow{R^{n}} $ defined by $L(X)=AX$ for $X$ in $R^{n}$, then A is diagonliazable with $n$ linearly independent eigenvectors ...
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45 views

Constructing a bilinear form on $\mathbb{R}^2$ that gives rise to a particular matrix

As the title says, I'm trying to create a bilinear form $B(\cdot, \cdot)$ on $\mathbb{R}^2$ with some particular constraints (which I do not know as yet) related to the Lorentzian space ...
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143 views

How can one prove the existence and uniqueness of solutions to linear differential equations?

It is a theorem (I think) that the equation: $$\mathbf{x}'(t) = A(t)\mathbf{x}(t) + \mathbf{b}(t); \qquad \qquad \mathbf{x}(t_0) = \mathbf{x}_0$$ Has a unique global solution for any matrix ...
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84 views

Finding fundamental solution to x^2 - 2y^2 = 1 using a straight line

I am trying to solve $x^2 - 2y^2 = 1$ by introducing Line $L$ with $y = m(x-1)$ as we know that one of the solution's is at $(1;0)$ I get the equation $(1-2m^2)x^2 + 4m^2x - (2m^2+1)$ and then divide ...
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37 views

Linear Transformations with polynomials

Hello everyone, I'm just confused as to wear to start with this question. Should I create a matrix from the polynomials given to me before the equals sign or after the equals sign? Is that the ...
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87 views

Finding the closest point between a plane formed by a equation and another point

There's a question about the geometry on the $ \mathbb{R}^{3} $ topic of my book that goes as follows: Consider the plane $\pi$ : $ 2x + 3y - z = 20 $ and the point $P $(-4 ,2 ,6). Find the point ...
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21 views

Find a Basis of the Image of $A$

I'm having trouble solving this equation. I know how to find a basis for the null space but I'm unsure how to find the basis for the image of A. Any help on how I approach this problem?
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108 views

What Do Hilbert Spaces Look Like?

For any vector space $V$ over $\mathbb{C}$, let $X$ be a set whose cardinality is the dimension of $V$. Then $V \cong \bigoplus\limits_{i \in X} \mathbb{C}$ as vector spaces. Is there a similar ...
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85 views

Proof of the equivalent properties of inner direct sum

So the question is the following: Prove the following are equivalent: $V$ is the inner direct sum of the subspaces $U_1,U_2,\dots ,U_n$. $V = U_1+\dots+U_n$ and $\dim(V) = ...
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19 views

Splitting groups of rows containing common columns in a binary matrix

So I've got a binary matrix. I'm looking to group the matrix's rows so that, in each set, each row will have at least one column where another row also has a $1 $ in that column. For instance, say ...
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65 views

Determinant of the sum of some special matrix

$A,B$ are $3\times 3$ matrices. It is known that: $\det(A)=0$ $\forall i,j: b_{ij}=1$, where $b_{ij}$ is an element of matrix $B$ $\det(A+B)=1$ Find $\det(A+2014B)$ I don't know what to do. I ...
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24 views

Prove that eigenvalues of $A$ are greater than those of $B$ iff $w_a>w_e$

$\newcommand{\H}{{\overline{H}}}$ Let $P \in \{0,1\}^{n \times k}$ such that $\forall_i\sum_{j=1}^k P_{ij}=1$ (one $1$ per row). Let $W = w_a(PP^\top - I) + w_e(11^\top - PP^\top)$. Basically, $W$ is ...
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44 views

Existence of a $p$-th root for an invertible matrix

Let $p$ be a positive integer and $M\in Gl_n(\bf C)$ Find some $A\in M_n(\bf C)$ such that $A^p=M$ This problem got me stumped. I can't even deal with $p=2$ (except when $M$ is a positive ...
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16 views

Computing S*S' and comparing to eye(5)

In matlab, we asked to set A=rand(5,2)*rand(2,5) then to set Q=orth(A), W=null(A'), ...
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29 views

Fundamental Subspaces: Orthonormal Bases

In matlab, we are asked to set A=rand(5,2)*rand(2,5) then to set Q=orth(A), w=null(A'), S=[Q W] the matrix S should be orthogonal. Why? (I have no clue on how to answer this)
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Matrix properties invariant under scalar multiplication

Given a square real matrix $A\in M_n(\Bbb R)$, what are ALL the properties invariant under scalar multiplication? In other words: which are the properties shared by all the $\lambda A$'s when ...
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66 views

Jacobian matrix dimension problem?

Assuming $g(x):R^n\rightarrow R^m$, where $x\in R^n$, does the following equality hold? $\dfrac{dg(x)^T}{dx}=\dfrac{dg(x)}{dx}=\dfrac{dg(x)^T}{dx^T}=\dfrac{dg(x)}{dx^T}$ I am confused about the ...
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20 views

Why does $A=UDU^H$ = $\lambda_1 u_1 u_1^H + \lambda_2 u_2 u_2^H$?

Just trying to figure out why this is true: $$A=UDU^H \quad\Leftrightarrow\quad A= \lambda_1u_1u_1^H + \lambda_2u_2u_2^H$$ $U$ is a unitary matrix composed of the eigenvectors of hermitian matrix A. ...
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18 views

Impulse mechanics issue

I dont see what I have done wrong with my arithmetic to get two different answers two different ways for this mechanics question? I used $I=mv-mu$ but get 6.5ms for one answer and 3.5ms for the ...
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46 views

Solution for $\mathrm{e}^{aX}=b I$ besides $X \propto I$?

Is there any solution of the equation $\mathrm{e}^{aX}=b I$, where $a$ and $b$ are numbers, $X$ is a matrix and $I$ is the identity matrix. One solution is of course $X=cI$, with some number $c$, ...
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need an example for better understanding

Two matrix system: Num1 and Num2 both have same left side(A), equal number of m and n, but the ...
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71 views

vector algebra proof

I've been working on the following question If ${\bf p}$, ${\bf q}$, and ${\bf r}$ are any vectors, demonstrate that ${\bf a}={\bf q} + \lambda\,{\bf r}$, ${\bf b} = {\bf r}+\mu\,{\bf p}$, and ...
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127 views

Proving a linear transform defined by an integral is injective

Let the fact that $I(p)(x)=\int_0^x p(s) ds$ is a linear transform from $P_4\rightarrow P_5$ be given. Prove that $I$ is injective. Would it be sufficient to just state that for any 2 ...
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47 views

Find the square of a 3x3 matrix that equals the negative Identity3 matrix

There are $2\times2$ matrices $A$ such that $A^2 = -I_2$. For example, if $$ A = \left [ \begin{array}{cc} 0& -1\\ 1 & 0\\ \end{array} \right ] $$ then $A^2 = -I_2 = \left [ ...
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63 views

What are the analogs of quadratic forms of degree $k>2$?

Let $f:{\Bbb R}^n\to{\Bbb R}$ be a smooth function. Its differential of $k$-th degree in a point $a\in{\Bbb R}^n$ can be defined as a map $$ d^k f(a):{\Bbb R}^n\to{\Bbb R},\qquad ...
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45 views

Find Jordan canonical form with Kronecker product

Let $f: K^3\to K^3$ be a map in Jordan canonical form having a matrix $$\begin{pmatrix} 1 & 1 & 0\\ 1 &0&1\\ 0&1&1\\ \end{pmatrix}$$ Find the JCF of the map $f\otimes f$. My ...
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32 views

classification of Jordan forms

A real $8\times 8$ matrix $A$ has $2-i$ and $3+4i$ among its eigenvalues, and their algebraic multiplicity is 2. Write down the possible generalized (real) Jordan matrices for $A$. How can I use the ...
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99 views

$\mathbb{R}^n$ contains at least two subspaces. True or false?

I was helping a friend with her linear algebra homework and I was wondering if my response for this question is right and a more elaborate answer of this question because I feel like I myself don't ...
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33 views

find special basis to make null space part equal to zero

Let $V,W$ be two vector spaces over a field $K$. Assume $dim(V)=m,dim(W)=n$. Consider the pairs $(A,w)$ where $A:V\to W$ is a linear map and $w\in W$. (a) Assume that $w\notin Im(A)$. Prove that ...
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Question on similar matrices and diagonalization.

Let $A$ and $B$ be two $n\times n$ matrices with entries in a field $k$. If $A$ and $B$ are similar, then can each one be obtained from the other via a combination of elementary row and column ...
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45 views

Proving a linear transform is injective

Let $A:V \rightarrow W$ be a linear map. Prove that A is injective iff $\{v \in V :Av=0\}=\{0\}$ I read that a linear transform is injective iff the kernel of the function ...
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128 views

Is a complex vector space closed under complex conjugation?

Given a complex vector space $\mathcal{V}$, its complex conjugate $\overline{\mathcal{V}} = \{ \overline{v} : v \in \mathcal{V} \}$ consists of the "same" set of points (according to a number of ...
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32 views

Show that $U$ is both unitary and hermitian

If $u$ is a unit vector in $\mathbb{C}^n$, and $U=I-2uu^H$, show that $U$ is both unitary and hermitian and that therefore it is it's own inverse. My attempt: Since $u$ is a unit vector it's norm ...
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45 views

Checking if the set of solutions to $y''+ 4y'+ 8y = 0$ is a vector space with the usual operations.

I've read a question that ask me to check that the set of functions on the line that has second derivative and verify the equation $y''+ 4y'+ 8y = 0$ is a vector space with the usual operations. I ...
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1answer
46 views

Proof of linear independent eigenvectors

Hello I am looking for insight onto this theorem I will post. I will also post what I have for a potential proof, but I don't think it is very rigours. I am looking for one that maybe uses induction ...
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21 views

How to write the equation of a graph which has two or more different functions togather in that graph?

Like in the graph below the velocity is linear for some time and then becomes exponential. We know that the equation of the straight line is V(t) = m (t-h)+c and the equation of the exponential ...
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43 views

Orthogonal Complex Matrices

Consider $M_{2\times2}(\Bbb C)$ together with the inner product $<A,B>=Trace(B^\dagger A)$, where $B^\dagger$ is conjugated. Let $W$ be the subspace defined by $W= \left ...
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88 views

Proof Verification of Cayley Hamilton

I am wondering about how this proof I am doing for the cayley hamilton is and if it is fully valid. I am also interested in any suggestions, better options or things I should note. Or if I am on right ...
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3answers
86 views

Prove there are no other invariant subspaces

Let $f \in End(V)$ has $n\times n$ matrix at basis $v_1, … ,v_n$ which is jordan block($n \times n$) ...
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Properties of eigenvectors of a sample covariance matrix?

My apology if the question is not appropriate. For me Eigenvectors are quite a mystery. Does it have any property that we can relate to the matrix it came from? By property I mean something like the ...
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58 views

Why the principal components correspond to the eigenvalues?

Suppose ${\bf{X}} = ({X_1},{X_2},\ldots,{X_n})$ are the original components (also random variables) and ${{\bf{w}}_j} = ({\omega _1},{\omega _2},\ldots,{\omega _n})$ are loadings for the $j$th ...
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Shamir's secret sharing interpolation problem

I try to understand this protocol - Shamir's secret sharing - threshold scheme. I got my data and I made interpolation basing on examples published on Wikipedia. You can see them below (sorry, I am ...
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56 views

It is possible to orthogonalize a set of linearly independent vectors via SVD?

Let's say I have a set of linearly independent vectors, collected in a square matrix $\mathbf{M}$. I know that I could orthogonalize these vectors with the QR decomposition, $\mathbf{M} = ...
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53 views

Basis of a basis-linear algebra?

Usually when we say that $v_1$ and $v_2$ are basis we imply that they are linearly independent and span the space. We by default denote $v_1$ and $v_2$ in $i$-$j$ basis. Then how is $i$-$j$ basis ...
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40 views

Find linear transformation such that N(T)=R(T)

Well the title quite says it all, only I must say $T:\mathbb{R}^{2} \mapsto \mathbb{R}^{2}$. I understand that $N(T)=\{v:T(v)=0\}=\{T(v):v\in \mathbb{R}^{2}\}=R(T)$ implies that $T(T(v))=0$ for any $v ...
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Find the closest element in a span of matrices?

Given a span: $$\left\{\begin{pmatrix} 1 & 0\\ 0 & 1\end{pmatrix}, \begin{pmatrix} 0 & 1\\ -1 & 0\end{pmatrix}\right\}$$ find the closest element to ...
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60 views

Finding basis for eigenspace when RREF returns several non-zero rows

Given a matrix A: $ \left(\begin{array}{rrr} 1 & 1 & 3\\ 1 & 3 & 1\\ 3 & 1 & 1 \end{array}\right). $ The eigenvalues are 5, 2 and -2. Now I have trouble with the eigenvalue ...
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82 views

Proving $ax+b$ is a linear function

$L\colon\mathbf{R}\to \mathbf{R}$ be given by $L(x)=ax+b$ over the scalar field $R$. I understand for that a function to be linear, it must adhere to the properties of additivity and scalar ...
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37 views

Node Game Recursion Problem

http://i.imgur.com/LwNr4rn.png I'm trying to figure out part a. However, I'm not sure if the set of simultaneous equations I've found is correct. Or at least, I can't solve the set. Any help would be ...