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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151 views

Flipping a vector across the y-axis

Say I have a vector A=[2,2] and I want to express it as [-2,2] (pretending I don´t know the coordinates). Notice that this is the same vector flipped over the y axis...How do I do this? A negative ...
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0answers
49 views

Producing integer combinations of irrational numbers in sequence?

Let $\mathbf{w}=\{w_0,w_1,\cdots,w_n\}$, $\mathbf{k}_i=\{k_0^i,k_1^i,\cdots,k_n^i\}$ and $\mathbf{m}_i=\{m_0^i,m_1^i,\cdots,m_n^i\}$, where $w_j\in\mathbb{R}$, $k_j^i\in\mathbb{Z}$ and ...
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1answer
107 views

moore-penrose inverse of complex square matrices

How can we find the moore penrose inverse of a complex square matrix? Can you give me an example?Actually i need a concrete and detailed example. so please help me.Thank you
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2answers
128 views

Logarithm of matrix with positive entries

For matrices with positive entries (or more generally, irreducible matrices with non-negative entries), we have the Perron-Frobenius theorem, which tells us that there will be a unique eigenvector ...
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1answer
277 views

Gauss-Jordan Elimination to solve for variables

I have the following linear system: $$x + 2y - 3z = 4$$ $$3x - y + 5z = 2$$ $$4x + y + (s^2 - 14)z = s+2$$ Im trying to solve for $s$ to figure out how many solutions it has (if any). I know how to ...
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3answers
158 views

Matrices - Find the rank and determine if its invertible

Find the rank of $A = \begin{bmatrix}2&1&-4\\-4&-1&-6\\-2&2&-2\end{bmatrix}$ and explain why $A$ is not invertible. What I have done is: Guass-Jordan Elimination: ...
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1answer
42 views

Matrices - Find matrix E

Suppose $A = \begin{bmatrix}1&2&-1\\1&1&1\\1&-1&0\end{bmatrix}$ and $D = \begin{bmatrix}1&2&-1\\-3&-1&3\\2&1&-1\end{bmatrix}$. I need to find the matrix ...
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513 views

Write the following in the form of AX = B

Write the following system of equations in the form $AX = B$, and calculate the solution using the equation $X = A^{-1}B$. $$2x - 3y = - 1$$ $$-5x +5y = 20$$ I'm not the strongest at linear algebra ...
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1answer
31 views

normalize a vector in $\mathbb C^3$ - a very basic question

I think I forgot a bit previous-year Linear Algebra, so I have a very basic question to you. Given the following question: Normalize the following vector: $v \in \mathbb {C^3}, \space v = i, -i, ...
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30 views

I need some help regarding definite matrix

Is a non-negative definite matrix a positive definite matrix?
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1answer
160 views

Solve system of ODEs

I have a system of related ordinary differential equations (ODEs) that look like: $$C x'(t) + Fx''(t) = 0$$ where $x$ is a $n$-dimensional vector, and $C$ and $F$ are square $n \times n$ ...
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1answer
33 views

A problem similar to matrix scaling

I'm interested in solutions to the following problem, which is clearly related to the problems known as matrix scaling and matrix balancing (as described in (1), for example), but is different from ...
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3answers
548 views

Prove that the distance between parallel planes $\vec{n}\cdot \vec{x} = d_1 $, $\vec{n}\cdot \vec{x}=d_2$ is $|d1-d2|/||\vec{n}||$

Prove that the distance between parallel planes with equations $\vec{n}\cdot \vec{x} = d_1 $ and $\vec{n}\cdot \vec{x}=d_2$ is given by ...
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2answers
366 views

degree of homogeneity

I have the function $$f(x,y)=\frac{y^b}{x^a}+\frac{x^b}{y^a}\quad a,b\gt0$$ The questions I have to answer are For which a and b is the function homogenous? Determine the degree of homogeneity My ...
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1answer
224 views

Alternating multilinear map and products

I was reviewing some school notes from many semesters ago and I came across a point which I wish to prove but can't. Let $F$ be a field (real or complex for example), and we say $\delta : ...
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1answer
48 views

A question about diagonalizable.

For which $x$ is $$M=\begin{pmatrix}4&0&-2\\x&5&4\\0&0&5\end{pmatrix}$$ diagonalizable? I know a matrix which is diagonalizable can be written in the form $A=S\Lambda S^{-1}$ ...
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1answer
30 views

Find a matrix A

Find a matrix $A$ such that $(A - 3\mathcal{I}_2)^{-1}$ = $\begin{bmatrix}1&2\\3&4\end{bmatrix}$ I dont understand what the question is asking and how to solve it! Any ideas?
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3answers
212 views

Matrices - Find the value(s) of constant k

Find the values of the constant $k$ such that $(k$A$)^T(k$A$) = 28$, where: $$A = \begin{bmatrix}-1\\2\\-3\end{bmatrix}$$ Actually, I got no idea how to solve this. how do i solve this? Can you ...
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2answers
21 views

Matrices satisfying the following relation

Find complex matrices $M_1, M_2, M_3$ such that $$M_i M_j + M_j M_i = 0$$ for $i \neq j$, and $M_i^2 = I$. I am stuck. Is the expression $AB + BA$ called something? It's sort of like a commutator.
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55 views

Matrices - Prove A and B are symmetric 2 x 2

Prove that if $A$ and $B$ are both symmetric 2x2 matrices, then $A$ + $2B$ is also a symmetric matrix. The problem I have with this question is proving. How do I prove that? All I understand to do is ...
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2answers
78 views

$n^{th}$ root of a matrix.

What conditions do I need on a matrix $A$ in order to know an $n^{th}$ root exists. In other words there is a matrix $B$ such that $B^n=A$ for $n \in \mathbb{Z}^+$.
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2answers
106 views

Matrices - Find x, y and z

I have two matrices $A$ and $B$ and I'm trying to figure out what $x$, $y$, and $z$ are. $$\begin{bmatrix}x+2y&x\\-x+y&2x-y\end{bmatrix} = ...
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2answers
450 views

Is the transpose of a linearly independent matrix linearly independent? [closed]

Question as above. Is the transpose of a linearly independent matrix linearly independent? Also does the product of two linearly independent matrices result in a linearly independent matrix?
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52 views

How to prove that the determinant is the same no matter how you take it?

To find the determinant, pick a row and move along it creating minors and use the recursive definition of determinant. How do we know that the determinant will be the same no matter which row you ...
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1answer
62 views

Finding the transformation matrix of this linear map.

I've being doing several exercises and none was of this kind, which I can't figure out: Let $V$ and $W$ be vector spaces with basis $B=\{\vec{v_1},\vec{v_2},\vec{v_3}\}$ and ...
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1answer
73 views

Jordan's decomposition

I have a matrix $A\in R^{n,n}$. $A= \begin{bmatrix} 1&0&-2&0&0&\dots&0\\ 0&1&0&-6&0&\dots&0\\ 0&0&1&0&-12&\dots&0\\ ...
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2answers
84 views

A question about diagonalizable matrices

Let $A$ be a square matrix such that $A \ne0$, but $A^k=0$ for some integer $k \gt1$. show that $A$ is not diagonalizable. Could somebody give me some hints?Many thanks
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0answers
56 views

A question in linear algebra.

Let $A$ be a symmetric matrix with real coefficients. If $A^n=I$ for some integer $n\ge3$, prove that $A^2=I$ I have no idea in proving this question, hope somebody can help me.
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5answers
5k views

How to tell if two 3D vectors are in the same direction?

Given: $$AB=\left( \begin{array}{ccc}2\\1\\3\end{array} \right) \;\;\;\; \text{and}\;\;\;\; CD=\left( \begin{array}{ccc}4\\3\\6\end{array} \right).$$ Justify if $AB$ has the same direction as ...
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2answers
44 views

Prove the following

Prove that if $p$ and $q$ are polynomials over the field $F$, then the degree of their sum is less than or equal to whichever polynomial's degree is larger $$\deg(p+q)\leq \max \left\{\deg(p),\deg(q) ...
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1answer
34 views

Solve a system of three equations by rewriting in row-echelon form

I tried solving this system of equations and I got what seems to be an inconsistent system. I wanted to post my results here to see if I'm correct. Here is the original problem: $$ \left\{ ...
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1answer
39 views

Fractional and irrational matrix powers

What is a good reference to learn the basics about raising a matrix to a rational power or an irrational power? So I am interested in the existence and computation of things like $A^{\frac{1}{3}}$ or ...
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1answer
33 views

Fixpoints of affine transformations

I want to find out all the possibilities what fixpoints of an affine transformation can be in 2-dim vector space. If the transformation is identity, then it is trivial - fixpoints describe the ...
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1answer
894 views

Get the rotation matrix from two vectors

Given $v=(2,3,4)^t$ and $w=(5,2,0)^t$, I want to calculate the rotation matrix (in the normal coordinate system given by orthonormal vectors $i,j$ and $k$) that projects $v$ to $w$ and to find out ...
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0answers
98 views

Reference for Codimension in Infinite Dimensional Normed vector spaces

There are a couple identities I would like to use related to the codimension and its relationship to the annihilator; some of these seem to be true for all normed vector spaces, and others seem only ...
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1answer
98 views

Equations of planes and lines in 3-space

I'm reading Strang's book "Linear Algebra and it's applications" and he writes in the first chapter that an equation involving two variables in still a plane in 3-space. "The second plane is 4u - 6v ...
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1answer
146 views

How to write a linear map as a matrix with respect to a given canonical basis

I am asked to write a linear map as a matrix with respect to a given canonical basis. The basis is $b = \left \{ \begin{bmatrix} 1 \\ 0 \end{bmatrix} , \begin{bmatrix} 0 \\ 1 \end{bmatrix} \right \} ...
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3answers
110 views

Solve this system by rewriting in row-echelon form $x+y+z=6$, $2x-y+z=3$, $3x-z=0$

This is my very first problem in Linear Algebra and I guess I really need to brush up on my Algebra skills..I'm at a loss as to how to solve this equation My reading said that there are basically 3 ...
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1answer
54 views

Is it true that positive definite matrices generates all the symmetric matrices?

Is it true that positive definite matrices generates all the symmetric matrices in $M_n(\mathbb{R})$? And is it true that the set of nonsingular symmetric matrices generates all the symmetric ...
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0answers
59 views

Verify: Orthogonal matrices are diagonalizable.

Verify: Orthogonal matrices are diagonalizable. I can't reach in any conclusion. All I can see is that an orthogonal matrix has $n$ linearly independent rows. However any nonsingular matrix has the ...
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1answer
22 views

Recovering $X,Y,Z$ from $x,y,z$ in CIE color model.

The background of where I'm getting this from is less important, but you can read it if you like. The question is, given three variables $X$, $Y$, and $Z$, all ranging from $0$ to $1$, we can ...
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1answer
103 views

Equation of a line through a point that intersects two crossing lines.

Find the equation of a line through a point, $P(7,1,1)$, that intersects two crossing lines $a$ and $b$. Where $$ a\;\begin{cases}2x+z&=0\\2x-y-1&=0\end{cases} \quad\text{ and }\quad ...
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0answers
89 views

Moore-Penrose pseudoinverse of a 3×3 matrix

Is there a "simple" formula for computing the Moore-Penrose pseudoinverse of a $3\times 3$ matrix? I mean something like the formula for the inverse (for non-singular matrices), which involves the ...
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1answer
258 views

Derivative of $(Ax - b)^T(Ax-b)$

I am trying to take the derivative of $(Ax - b)^T(Ax-b)$ and setting it to zero without expanding the multiplication, by only using matrix calculus. I knew the partial derivative of $x^Tx$ according ...
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1answer
91 views

Dual and Second Dual Basis

Let $B={e_1, e_2, e_3 }$ the canonical basis of $\mathbb{R}^3$. Build the dual and second dual basis of $\mathbb{R}^3$. This is a question about finding the base to a vector space which makes a ...
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1answer
89 views

Linear operator $T^k$ effect on $kerT^k$ and $ImT^k$

Let $T:V\rightarrow V$, an linear operator. In general, what can you say about $kerT^k$ and $ImT^k$? For example, I've understood that $kerT^{k-1} \subseteq kerT^k$. I'd like to know what else ...
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1answer
665 views

Given an implicit 3D plane, how do I find the orthogonal projection matrix - which projects any point - onto this plane?

The plane is given by the equation $Ax+By+Cz+d = 0$. Can you tell me how can I figure out the 4x4 matrix which orthogonally projects any point given by homogeneous coordinates onto this plane? I am ...
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37 views

Intuitively what is it if making a modification of a torus?

It is well-known that if we have a equivalence relation in $\mathbb{R}^2$:$(z_1,z_2)\sim (z_1',z_2')$ iff $$\begin{pmatrix} z_1'\\ z_2' \\ \end{pmatrix}=\begin{pmatrix} 1&0\\ 0&1 \\ ...
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1answer
66 views

How to prove this by mean value theorem? $f(y)=f(x)+\nabla f(x)^T(y-x)+\frac{1}{2}(y-x)^T\nabla^2f(x+a(y-x))(y-x)$

How to prove this by mean value theorem? $f(y)=f(x)+\nabla f(x)^T(y-x)+\frac{1}{2}(y-x)^T\nabla^2f(x+a(y-x))(y-x)$ where $a\in[0,1]$. The mean-value theorem is $\frac{f(y)-f(x)}{y-x}=\nabla ...
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2answers
38 views

Does conjugation preserve spectrum of matrices?

Actually, I saw normalizer of diagonal matrices are permutation matrices. I read the answer but I don't know how to prove that conjugation preserves the spectrum. Actually I do some proof on 2x2 ...