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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question about dual space and linear form

I stuck at the following linear algebra problem. Could you give me some hints? Let $V$ be a vector space. Given $g,\,f_1,\,f_2,\,...,f_r\in V^*$, prove that $g\in ...
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18 views

Correct proof relating to diagonalizability

Regarding the question Prove that if $B$ is diagonalzable then $A=0$ when $$B=\left(\begin{array}{cc} A & A\\ 0 & A \end{array}\right)$$ I have come up with the following proof: First ...
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6 views

Proof of distributive property of linear operator?

How can I show that for 2 linear operators $L$ and $M$ that transform some object $O$ into another object of the same type: $$(L(O)+M(O))*(L(O)+M(O)) = L(O)*L(O) + 2L(O)*M(O) + M(O)*M(O)$$ where $*$ ...
3
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3answers
27 views

Linear transformation onto and one to one?

(1)If a linear transformation $T:\mathbb{R}^n\rightarrow\mathbb{R^m}$ maps $\mathbb{R}^n$ onto $\mathbb{R^m}$ what is the relation between m and n? (2)If T is one to one what is the relationship ...
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7 views

Solve set of poorly conditioned linear equations in block matrix form

I would like to solve the following set of linear equations where A, B, C and D are each 4x4 matrices. K is then an 8x8 matrix The values in A and D have magnitudes of $\approx 10^{17}$, B has ...
3
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2answers
52 views

Proof that $-v = (-1)*v$

I need to prove that for every Vector Space this is valid: $$ -v = (-1)*v $$ -v = inverse element of addition -1 a real number $*$ the multiplication by real number of the Vector Space My teacher ...
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10 views

Relation between Discrete Sums in to Definite Integrals

$\sum_{r=i}^n \binom{n} {r} p^r (1-p)^{n-r}$ = $\int_0^p \frac{n!}{(i - 1)! (n - i)!} t^{i - 1} (1 - t)^{n - i} dt $ How can we verify this ? I have tried using integration on RHS but unsuccessful ...
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3answers
22 views

Expressing a matrix of rank $k$ as the sum of $k$ matrices of rank 1.

It's a well known fact that for any matrix $A \in R^{m \times n}$ which is the sum of $k$ matrices $A_1,...,A_k\in R^{m\times n}$ of rank $1$, it holds that $\operatorname{rank}(A) \le k$. My ...
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19 views

Non-negative Matrix [on hold]

Let $M \in \mathbb{R}^{n \times n}$ be a Metzler matrix, i.e., $M_{i,j} \geq 0$ for all $i \neq j$. Consider $c \in \mathbb{R}$ and $x \in \mathbb{R}^n$. Prove that the matrix $$ \text{diag}( x_1, ...
2
votes
2answers
40 views

Why is slope rise/run?

What makes slope rise over run? What makes the standard equation for a line use a slope of rise over run as opposed to run over rise? What would the standard equation of a line look like if m was ...
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1answer
21 views

Alternative ways to prove an easy set relation

I have a simple set relation, which is almost trivial to prove, but surprisingly, I can only prove it with an "indirect" method, which is bugging me: Let the set $L$ be a subset of $\mathbb{R}^n$. ...
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24 views

Prove that $dim(V)$ is an even number

Let $V$ vector space so that $dim(V)=n$ and let $T:V\to V$ a linear transformation so that $Im(T)=N(T)$. Prove that $dim(V)$ is an even number I have no idea how to star the problem. Can you give a ...
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0answers
10 views

4th order tensors and 2nd order matrices: operations are the same?

Working in the field of Computational Mechanics, it is frequent to encounter 4th order tensors that are transformed to 2nd order ones, or better in matrices (see Voigt's or Mandel notation). To do ...
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0answers
14 views

Kalman Filter for SLAM (robotics)

First, I want you to know that I'm a big mathematics noob. I'm currently working on a SLAM (Simultaneous Localization And Mapping) problem and when I browsed the Internet, I read about the Kalman ...
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1answer
20 views

Prove that every element of $V$ can be expressed as $w+cv_0$ for some $w\in \ker(T)$ and $c\in \mathbb R$

Let $V$ be a vector space over $\mathbb R$ and let $T:V\to \mathbb R$ a linear transformation. Suppose $\ker(T)\neq V$ and let $v_0\in V$ so that $v_0\notin \ker(T)$. Prove that every element of $V$ ...
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25 views

How to show $\exp(tX)\exp(tY)=\exp(t(X+Y)+tR(t))$ with $\displaystyle \lim_{t\to 0} R(t)=0$?

Let $X\in GL(n, \mathbb R)$. The exponential of $X$ is the matrix given by $$\exp(X)=\sum_{n=0}^\infty \frac{X^n}{n!}.$$ I need some help for showing the following result: ...
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12 views

Write down a linear programming problem

I want to replicate a linear programming problem.I have the following information, for the background." A fuzzy regression analysis with only one independent variable X results in the following ...
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0answers
18 views

Taking powers of a Vandermonde matrix elementwisely

Consider a Vandermonde matrix $$ V (x_1, x_2, \ldots , x_n) =\begin{pmatrix} 1 & x_1 & x_1^2 & \cdots & x_1^{n-1} \\ 1 & x_2 & x_2^2 & \cdots & ...
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1answer
27 views

Is $D_1+D_2P$ invertible when $P$ is positive definite and $D_1$, $D_2$ are diagonal matrices?

I came across this problem when proving another proposition: I have $A=D_1+D_2P$ where $P$ is a symmetric, real-valued, positive definite matrix and $D_1$ and $D_2$ are diagonal matrices with ...
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2answers
18 views

Constructing the matrix of the differentiation linear operator

For every integer $n > 1$ choose a basis in the space $P_n(F)$ of polynomials of degree not bigger than $n$, and construct the matrix of the differentiation linear operator $P_n(F)\longrightarrow ...
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2answers
113 views

A basis for a plane containing the origin.

I need to show that the set $\{(1,-1,0),(0,1,-1)\}$ forms a basis for the subspace $V\subset \mathbb{R}^3$ wherein $V=\{(x,y,z)\in\mathbb{R}^3 \ \vert \ x+y+z=0\}.$ I tried to show that from the ...
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votes
1answer
11 views

How to determine whether point is on the left of right side of the line

I have a line, for example $y=-\frac{1}{9}(x+6)-3.5$. And I also have a point, for example $A(-5, 2)$. How to determine whether this point is on the line, on the right or on th left side? Just put A ...
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2answers
43 views

Finding the eigenvalues of a given Markov matrix

Let $$A = \begin{pmatrix} 0.6 & 0.1 & 0.1\\ 0.1 & 0.8 & 0.2\\ 0.3 & 0.1 & 0.7 \end{pmatrix}$$ I want to find the eigenvalues of this matrix. Because this is a markov matrix, ...
2
votes
3answers
46 views

To prove that $A$ has a one-dimensional eigenspace , where $A \in SO(3)$ , $A \ne I$

Let $A\ne I$ be a $3\times3$ real orthogonal matrix with determinant $1$ , then how to prove that $A$ has a one-dimensional eigenspace ?
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1answer
74 views

$f(AB)=f(A)f(B)$, show that $f$ is or injective or zero

Let $f\in\mathcal{L}(\mathcal{M}_n(\mathbb{R}))$ such that: $\forall(A,B)\in\mathcal{M}_n(\mathbb{R}),f(AB)=f(A)f(B)$ How can I show that $f$ is or injective or the null function ? What I have ...
1
vote
1answer
79 views

Prove that if $B$ is diagonalzable then $A=0$

Let $A$ be a real $n\times n$ matrix and $B$ be a $2n\times 2n$ matrix such that: $$B=\left(\begin{array}{cc} A & A\\ 0 & A\\ \end{array} \right)$$ Prove that if $B$ is diagonalizable then ...
2
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0answers
17 views

Skew symmetric matrices even size commutativity

Given Data in the question $w(t)=\frac{1}{2}\begin{bmatrix} 0 &r(t) &-q(t) &p(t) \\ -r(t)& 0 &p(t) &q(t) \\ q(t)& -p(t) &0 & r(t)\\ -p(t)&-q(t) ...
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21 views

Is this correct? A snail climbs up a 20 metre wall. [duplicate]

A snail climbs up a 20 metre wall. Every hour it climbs up 5 metres then slides back 3 metres. How long does it take the snail to climb up the wall. My solution: By 16 m, the snail would have taken 8 ...
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19 views

reflect a point about a plane using matrix transformation

Given the plane equation ax + by + cz = d and an arbitrary point in 3D space (i, j, k), how do i find a homogenous transformation matrix that is used to reflect this point about the plane?
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3answers
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Maths problem on distance and time. [on hold]

A snail climbs up a 20 m wall 5 m every hour then slides back 3 m. How long does it take the snail to climb up the wall? Possible solutions and working out please.
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finding the range of the linear transformation

If $T: P\to P $ defined by $T(p)(x)=p^"(x)-2p(x)$ then how shall I find the range of $T$ . The answer is given but I don't know the process to finding out . So please explain.
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1answer
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If $X$ and $Y$ are subspaces of $\mathbb{R^n}$, is $X+Y$ a subspace of $\mathbb{R^n}$?

Here we define $X+Y$ as the set of all vectors in $\mathbb{R^n}$ of the form $x+y$ where $x \in X$ and $y \in Y$. So suppose that $X$ and $Y$ are two distinct lines in $\mathbb{R_3}$. Then $X+Y$ is ...
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1answer
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Let $Z\subset X$ and $x_0\in X-Z$. Show that there is a linear functional $f$ on $X$ such that $f(x_0)=1$ and $f(x)=0$ for all $x\in Z$.

Let $Z$ be a proper subset of an $n$-dimensional vector space $X$, and let $x_0\in (X-Z)$. Show that there is a linear functional $f$ on $X$ such that $f(x_0)=1$ and $f(x)=0$ for all $x\in Z$. I ...
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1answer
13 views

Deconstructing the basis of a matrix?

So I was given this matrix and was asked to find a basis for it. $\begin{pmatrix} a & a+b\\ c & d \end{pmatrix}$ After some looking I realized that it was the matrices \begin{pmatrix} 1 ...
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1answer
16 views

how can i find the inverse transformation of the following transformation?

if $ T: P_2 \to P_2$ defined by $T(\alpha_0 +\alpha_1 x+\alpha_2 x^2)=(\alpha_0 +\alpha_1)+(\alpha_1+2\alpha_2)x +(\alpha_0+\alpha_1+3\alpha_2)x^2$ then how shall I find the inverse $T^{-1}$
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337 views

How to prove that a quadratic equation implies both variables are zero

This might be really simple, but I can't find how to prove that $a^2-\frac{2}{3} ab+b^2=0$ implies that both $a$ and $b$ are zero. Any help will be appreciated!
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32 views

True or False Linear Algebra-Subspace [on hold]

I need some help justifying why these are either true or false A. If A is a $ 5 \times 3$ matrix, then $null(A)$ forms a subspace of $\mathbb{R}^5$ B. The set of integers forms a subspace of ...
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1answer
24 views

What do I do with these equations to create a Jacobian matrix?

My instructions were to find equilibrium values (the picture I added is only showing E0, I was hoping if I got it figured out I could do the others rather than someone try to do all of them for me), ...
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How to find LU factors of a matrix when diagonals are changed

Say I have $A=LU$ already factored into lower and upper triangular matrices $L$ and $U$. Now I want to work on the eigenvalue problem $A-\lambda I=A'=L'U'$ where prime indicates new matrices. Given ...
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40 views

find eigenvalues and eigenvectors of given operator

find eigenvalues and eigenvectors of the following operator A: $$A~v_1 = v_2 \text{ and } A~v_2 = v_1$$ I tried the following: $$A~(v_1+v_2) = A(v_1)+A(v_2) = v_1+v_2$$ so eigenvalue is $1$ and ...
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1answer
17 views

Use the pseudoinverse to find the conic section of best fit to the data

I am working on a group project and none of us can figure out how to find the answer. Our professor insists that all of our work be done in maple. The problem is: Use the pseudoinverse to find the ...
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26 views

Algebra derivation

I am deriving Lagrange multipliers for Support Vector machine algorithms. $$L(\omega,b,\alpha)=\frac{1}{2}\|\omega\|^2 - \sum\limits_{i=1}^{m}\alpha_i[y^{(i)}(\omega^Tx^{(i)}+b)-1]$$ where $x^{(i)}, ...
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3answers
44 views

What is the relationship between $\ker(A), \ker(A^2), \ldots, \ker(A^n)$?

If $\vec{y} \in \ker(A)$ then $A\vec{y} = \vec{0}$ so that $(A*A)\vec{y}= A*(A\vec{y})=\vec{0}$ and $\vec{y} \in \ker(A*A)$. If $\vec{y} \in \ker(A*A*\cdots *A)$, then $(A*A*\cdots ...
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1answer
21 views

distance from intersection of linear subspaces

Consider two points $x, y \in \mathbb{R}^n$, two linear subspaces $V_1, V_2 $ of $\mathbb{R}^n$ and a distance $d(\cdot , \cdot)$ in $\mathbb{R}^n$. Assume that $d(x,V_1) \leq d(y,V_1)$ and $d(x,V_2) ...
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2answers
20 views

Find a linear transformation whose kernel is the line spanned by $(-1,1,2)$ in $\mathbb{R^3}$

I tried to reason about this problem using the $RREF$ of the matrix of the linear transformation. Because I know that if I can get the $RREF$ to show homogeneous equations: $x_1 = -x_2$ $x_2: free$ ...
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1answer
21 views

Question on the set of $A\in\mathcal{M}_n(\mathbb{R})$ so that there exists $r\in\mathbb{R}$ such that $^tAA=rI_n$

Let $C_n$ be the set of $A\in\mathcal{M}_n(\mathbb{R})$ so that there exists $r\in\mathbb{R}$ such that $^tAA=rI_n$. Let $d_n$ be the maximal dimension of a vector subspace of ...
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0answers
17 views

Properties of $Hom(V,W)$ for $\theta \in Hom(V,W)$ and $\theta^T \in Hom(W^*,V^*)$

Properties of $Hom(V,W)$ for $\theta \in Hom(V,W)$ and $\theta^T \in Hom(W^*,V^*)$: Define $\theta^T: W^* \rightarrow V^*$ by $\theta^T(\tau)=\tau \theta$ for all $\tau \in W^*$ 1.) Prove that ...
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vote
1answer
21 views

Finding standard matrix linear transformation?

$T: R^2 \rightarrow R^2$ First rotate point through $\frac{-3\pi}{4}$ clockwise and the reflect points through the horizontal x axis. I know what $T(e_1)$ and $T(e_2)$ is Let $a= \begin{bmatrix} ...
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0answers
6 views

derivative of composition of rotations

Let $\theta$ and $\psi$ be two vectors in $\mathbb{R}^3$. I want to compute $$\nabla_{\psi} \log \left( e^{[\theta]_\times}e^{[\psi]_\times}\right)$$ Where $[v]_\times$ is the skew-symmetric ...
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1answer
15 views

Translating verbal descriptions into algebraic expressions

I'm trying to solve the following problem but the result I'm getting is not logical given the data of the problem. Pat invested a total of \$30,000. Part of the money was invested in a money ...