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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3
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1answer
69 views

Maximization of a determinant

I'd like to compute $$ \DeclareMathOperator*{\argmax}{arg\,max} A^*=\argmax_{\substack{A\in\mathbb{R}^{d\times k}\\A^T A=I}} \det(A^T \Lambda A) $$ where $k\leq d$, $\Lambda=\operatorname{diag}(\...
0
votes
1answer
1k views

How do you find the vector x determined by the given coordinate vector and given basis B?

I saw a couple different ways to approach this problem from tutorials on YouTube, and each led to a different answer. This is what I got: 3 -4 | 5 -5 6 | 3 3 * 5 + -4 * 3 = ...
0
votes
1answer
35 views

Notationquestion: Matrices

I have to determine the base change matrix $S=M(id,A,B)$. Now i looked at how to do it, but which base is right and which left in doing it? i would write A|B and then try to get the elementary ...
2
votes
1answer
32 views

If both of $A,A^{-1}$ have entries from non negative integers then can we say $A$ is a permutation matrix?

I've shown if both of $A,A^{-1}$ (assuming $A$ to be invertible) are $n\times n$ matrices with entries from natural numbers then both of them have to be permutation matrices. Now my question is if ...
-1
votes
2answers
41 views

Finding a point between 2 moving points colinearly , given 2 moving points and distance. [on hold]

A------B---------------C A and C are moving points that can move anywhere A = (xa, ya), C = (xc, yc) B (xb, yb) is a point between A and C colinearly With one condition that distance of AB = ...
9
votes
2answers
800 views

Why is some power of a permutation matrix always the identity?

If you take powers of a permutation, why is some $$ P^k = I $$ Find a 5 by 5 permutation $$ P $$ so that the smallest power to equal I is $$ P^6 = I $$ (This is a challenge question, Combine a 2 ...
0
votes
1answer
60 views
+50

Rank and null space of a particular block matrix.

Let $D_1, D_2 \in \mathbb{R}^{N \times N}$ be diagonal matrices with diagonals that are linearly independent vectors. Let $A, B \in \mathbb{R}^{N \times N}$ be rank-deficient matries. Define $S = \...
1
vote
1answer
10 views

Proof: $x_{1-n}$ linear dependant, w alternating multilinearform $\Rightarrow$ $w(x_1,…,x_n)=0$

Let $F$ be a field and $X$ a $F$-linear Space with $dim_FX=n\in\mathbb{N}$. Let $w$ be an alternating multilinearform on $X$ and let $x_1,\cdots ,x_n\in X$ be linear dependant. Show that $w(x_1,\...
1
vote
1answer
17 views

Eigenvalues of product of p.d. Matrix with upper-triangular Matrix

Let $A$ be a positive definite matrix (positive eigenvalues). Let $B$ be an upper triangular matrix, with ones in its main diagonal (i.e. all its eigenvalues are 1). Is there anything I can say about ...
4
votes
1answer
50 views

Prove a Hermite polynomial property by linear algebra

Let $$ X=\begin{pmatrix} 0 & & & &\\ 1 & 0 & & &\\ & 1 & 0 & &\\ & & \ddots & \ddots &\\ & & & 1 & 0 \end{pmatrix}, D=\...
0
votes
1answer
438 views

Edge weight function for graph instance of scheduling and allocation problem

I have difficulties developing a proper (non-scalar) edge cost function $c_e$ for my resource scheduling problem, which I mapped into a graph problem. Processes $P_i$ need resources $R_i \in \mathcal{...
0
votes
0answers
33 views

What is the Linear space $R^{2 \times 2}$ means visually in linear algebra?

Why mathematicians decided to define a space of matrices if it does not make any sense visually? What are the uses of such linear space? and why my professor tells me that matrices can be written ...
0
votes
1answer
26 views

Find $P$ such that $A=P^{-1}JP$ where $A$ is the matrix of $f$ and J is the Jordan Form. $P$ non invertible?

Find the Jordan Form and a basis of Jordan for the endomorphism of $R^4$ $$f(x,y,z,t)=(x,x+y-t,-2x+y+z+2t,-x+2t)$$ After doing all the process, I find $P$ such that $A=P^{-1}JP$ where $A$ is ...
-1
votes
2answers
42 views

If $A$ is diagonizable then $p(A)$ is diagonalizable

Show that if a matrix $A$ of size $n \times n$ is diagonalizable, then $p(A)$ is diagonalizable for each polynomial $p$.
0
votes
0answers
28 views

First steps in derivation of matrices spectrum

I was trying to go through a paper about 'The eigenvalue spectrum of a large symmetric random matrix' by Edwards and Jones (1976) and I found myself stuck at the very first step of a derivation. I ...
1
vote
2answers
74 views

what is the difference between cross product and exterior product?

I have learn that the exterior product is an oriented plane called bivector given as $A \times B = |A||B| \sin x (i \times j)$ For $x \in(-\pi,\pi)$. I will like someone to derive the cross product ...
0
votes
1answer
31 views

Transformations between coordinate frames

Suppose I have three coordinate frames: $A$, $B$ and $C$, all in 2D space. In homogeneous coordinates, I deduce, by inspection, the transformation matrices between each of these ($T_{AB}$, $T_{BC}$ ...
4
votes
1answer
2k views

Java Tetris - Using rotation matrix math to rotate piece

I'm working on building tetris now in Java and am at the point of rotations... I originally hardcoded all of the rotations, but found that linear algebra (matrix rotations) was the better way to go. ...
2
votes
1answer
41 views

Existence of a basis $B$ such that $M(\phi,B)=E$

Considering the matrix $$A=\begin{bmatrix}{2}&{1}&{0}\\{1}&{0}&{-1}\\{0}&{-1}&{-2}\end{bmatrix}\in M_3(R).$$ And $\phi:R^3 \times R^3\longrightarrow R$ the bilinear ...
2
votes
2answers
763 views

calculate centroid of triangle on a graph

Given ANY three points on a graph that form a triangle, how do you find the centroid using geometry? So basically I have three points (X1, Y1), (X2, Y2), and (X3, Y3). I am trying to use the slopes ...
2
votes
0answers
25 views

Action of the Product of Two Linear Functionals on a Polynomial

I am looking for help with the following problem. Here we denote the action of a linear functional on a polynomial by $$\langle L\mid p(x)\rangle$$ Suppose that there are two linear functionals $...
0
votes
1answer
28 views

The map $f$ is degenerate or non-degenerate?

Let denote by $M_{3,2}(\mathbb C) $ the space of all $(3\times2)$-matrix of complex-dimension equal $6$ with basis $(E_{1},E_{2},E_{3},E_{4},E_{5},E_{6})$. Let $f$ a $\mathbb R$-bilinear skew-...
0
votes
1answer
16 views

Orthonormal Basis /Linear Combination

(a) Find an orthonormal basis {${v_1,v_2,v_3}$} of the image of the linear function given by the matrix $$A=\begin{pmatrix} 1 & 1 & 2 \\-1 & 0 & 0 \\ -1 & 0 & 1\\ 0 & 1 &...
-3
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1answer
26 views

How to prove Binary field is a vector space? [on hold]

How can I prove that the binary field is a vector space over that field? Thank you
0
votes
1answer
29 views

Solving vector equation 2

Using vector method, show that the vector equation $$\bar{x}\times \bar{a}+(\bar{x}.\bar{b})\bar{c}=\bar{d}$$ is satisfied if $$\bar{x}=\lambda \bar{a}+\bar{a}\times \frac{\bar{a}\times (\bar{d}\...
3
votes
1answer
29 views

$p(x) \in \mathbb R[x]$ be a polynomial of odd degree , $n>1$ be an integer , then is the function $A \to p(A)$ surjective on $M(n,\mathbb R)$?

Let $p(x) \in \mathbb R[x]$ be a polynomial of odd degree , $n>1$ be an integer , then is the function $f: M(n,\mathbb R) \to M(n, \mathbb R)$ defined as $f(A)=p(A) , \forall A \in M(n,\mathbb R)$...
3
votes
1answer
40 views

$p(x) \in \mathbb R[x]$ be non-constant polynomial , $n>1$ , the function $A \to p(A)$ is surjective on $M(n, \mathbb C)$?

Let $p(x) \in \mathbb R[x]$ be a non-constant polynomial and $n>1$ , then is it true that the function $f:M(n,\mathbb C) \to M(n, \mathbb C)$ defined as $f(A)=p(A) , \forall A \in M(n, \mathbb C)...
0
votes
1answer
23 views

Zeros in pivot position

When zero appears in a pivot position, $$ A = LU $$ is not possible. What do we have to do here to make A=LU possible then? Do we have to find a specific P (permutation matrix) for A and continue ...
1
vote
2answers
52 views

Express the function $ f $ without using absolute value signs $\left|\frac{x-2}{x+3}\right|e^{\left|x-2\right|}$?

Good evening to everyone: This is the equation $$ f(x) = \left|\frac{x-2}{x+3}\right|e^{\left|x-2\right|} $$ What I've tried is: $$ \frac{x-2}{x+3}\ge 0 => x-2 \ge 0 => x \ge 2$$ Then $$ \frac{-...
7
votes
2answers
492 views

Additive rotation matrices

Let's assume that we want to find a rotation matrix which added to a given rotation matrix gives also a rotation matrix. I would name such matrix a rotation additive matrix for a given rotation ...
3
votes
1answer
50 views

Eiegenvalue equation

I have a matrix $M = D X X^T$, where $D$ is a diagonal matrix with real entries, and $X$ is a $n \times d$ matrix. Note that $M$ is not symmetric. I want to find the vectors $\alpha$ for which: $$X^T ...
3
votes
1answer
47 views

What is the name of this problem? linear Matrix equation optimization?!

I have almost no knowledge in linear algebra but I need to understand the process of solving a problem. In fact I'm looking for some keywords or hints to know what exactly should I be Googling! So any ...
0
votes
2answers
30 views

Solving vector equation 3

Solve for $\bar{x}$ and $\bar{y}$ $$\bar{x}+\bar{y}=\bar{a},~~ \bar{x}\times \bar{y}=\bar{b},~~ \bar{x}.\bar{a}=1$$ Attempt: $\bar{x}+\bar{y}=\bar{a}$ dot by $\bar{a}$, we get $1+\bar{a}.\bar{y}=|...
0
votes
1answer
22 views

Elimination and exchanging rows

Solve by elimination, exchanging rows when necessary $$ v + w = 0\\ u + v = 0\\ u + v + w = 1\\ $$ Which permutation matrix is required? answer is $$ P= \begin{bmatrix} 0 & 1 & 0 \\ 1 ...
0
votes
1answer
446 views

How do you calculate the dimensions of the null space and column space of the following matrix?

I understand you are supposed to get the reduced row echelon form, which I did, and this is what I came up with: 1 -2 0 19 -6 0 -37 0 0 1 -6 2 0 6 0 0 0 0 0 1 3 0 0 ...
0
votes
1answer
64 views

Why can't we sum two $n\times m$ and $u \times v$ matricies for all positive integer $n,m,u,v$? [on hold]

Why does the sum$$\left[\begin{matrix}1&2\\0&-1\\2 &3\end{matrix}\right]+\left[\begin{matrix}1&2&3&4\\0&-1 &1 &7\end{matrix}\right]$$ undefined? Let's expand these ...
0
votes
2answers
38 views

Give a geometric comparison of the solutions to [on hold]

Give a geometric comparison of the solutions to \begin{align} x_1 + 3x_2 - 5x_3 &= 4\\ x_1 +4x_2 -8x_3 &= 7\\ -3x_1 -7x_2 + 9x_3 &= -6 \end{align} and \begin{align} x_1 + ...
0
votes
2answers
80 views

Why can the equation Ax = b not be solved for every b

Let $A$ be a $3 \times 2$ matrix . Explain why the equation $A\vec{x} = \vec{b}$ cannot be solved for every $\vec{b}$ in $\mathbb{R}^3$. What about a $4 \times 3$ matrix? I'm not sure how to answer ...
0
votes
1answer
33 views

Showing sum of squared residuals is zero?

I have the model $$y_i = B_0+\sum\limits_{i=0}^pB_kX_{ik} + e_i$$ I'm looking to show the sum of squared residuals is zero if $p = (n-1)$. I have tried expanding it quite in depth and I haven't been ...
4
votes
2answers
14k views

Find a basis for a solution set of a linear system

I'm trying hard with this exercise but is breaking my back. Find a basis for the solution set of the given homogeneous linear system $3x_1+x_2+x_3=0$ $6x_1+2x_2+2x_3=0$ ...
3
votes
1answer
11k views

Determine a basis for the solution set of the homogeneous system

Determine a basis for the solution set of the homogeneous system: $$\begin{align*} x_1 +x_2 +x_3 &=0\\ 3x_1+3x_2+x_3 &=0\\ 4x_1+4x_2+2x_3&=0 \end{align*}$$ Then the augmented ...
0
votes
3answers
30 views

Evaluate integral over path using parametrisation

Evaluate the integral of $(F.dr)$ over the path using parametrisation [x=t, y=t, z=t] I know from simpler questions in class that you must find dx/dt, dy/dt and dz/dt but where do I go from there ...
-1
votes
1answer
53 views

Solving vector equation 1 [on hold]

Using vector method solve $p \bar{x}+\bar{x}(\bar{x}.\bar{b})=\bar{a}\times \bar{b}+\bar{c}$ How to solve $\bar{x}$ from such vector equation. Please help.
2
votes
2answers
614 views

Gram Determinant equals volume?

I have been trying to solve this problem of finding the 'n-volume' of a paralleletope spanned by m vectors, where clearly m =< n. In general, for computational purposes, what I have managed to do ...
0
votes
0answers
40 views

Counting GF($q=8$) matrices with a certain property

Let us denote by $\boldsymbol{v}_i$ the columns of an $m \times n$ GF($8$) matrix. The field elements are enumerated $\{0,1,2,...,q-1\}$. To define the arithmetic operations between field elements, we ...
2
votes
3answers
57 views

Check if a positive solution exist of a linear equation with two variables?

Let's say there's an equation $$a x + b y = c$$ where $a,b,c > 0$ are given. I want to know if positive solutions $x, y >0$ exist for this equation.
0
votes
2answers
34 views

Find a vector $w$ such that $Aw = v_1 + 3v_2$ [on hold]

if $A$ is a $3 \times 3$ matrix and $\vec{v_1},\vec{v_2},\vec{y_1},\vec{y_2}$ are vectors so that $A\vec{y_1} = \vec{v_1}$ and $A\vec{y_2} = \vec{v_2}$ find a vector $\vec{w}$ so that $A\vec{w} = \vec{...
0
votes
1answer
16 views

Last Step of a Parametric to Cartesian Conversion

I need to figure out how to combine the (4) line to make the t=y-z/R+x and I just don't have any ideas. I'm sorry if these seem basic but I'm 16 and struggling through a topic I've never done before. ...
0
votes
1answer
13 views

How to quickly find the basis of Null space in GF2

I have a matrix $A$ contains only 1 and 0. The operations is defined as in GF2, e.g., 1+0=1, 1+1=0, 0+0=0. I know how to make $A$ into row echelon form. For example, my $A$ now becomes ...
0
votes
1answer
23 views

Conditions for Non-negativity

Let's consider $A$ to be a square symmetric matrix whose entries are non-negative real numbers that sums to one. Even more, we shall consider its diagonal elements to be equal to zero. The question is:...