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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Matrix Algebra : Matrice * Transposed is a peculiar matrix

M is an n×n square matrix, containing only 0s and 1s and where each column sums to k, where k is a constant (say 5 in my case) and $~^tA.A=k.I + Ones$ (I is is the identity matrix and Ones in a n×n ...
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99 views

linear transformation mapping a rectangle to a parallelepiped?

I am puzzled by a statement in Billingsley's "Probability and Measure." After Theorem 12.2 (If $T:\Re^k\to \Re^k$ is linear and nonsingular, then $A \in \Re^k$ implies that $TA \in \Re^k$ and ...
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65 views

Direct sum of 2 subspaces to obtain $\mathbb{R}^4$

Consider $V$ as a subspace in $\mathbb{R}^4$ where $V= \text{span}\{(2,1,0,1),(2,-1,-1,-1),(3,0,2,3)\}$. Find a subspace $W$ so that $\mathbb{R}^4=W\oplus V$. If I find a vector in $W$ that is ...
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Can I replace the {1,3}-inverse by the Moore-Penrose inverse for Minimum Norm Solution?

Suppose the equations $Ax=b$ are compatible, and the Moore-Penrose inverse of $A$ is known as $A^{\dagger}$. In order to calculate the Minimum Norm Solution $$x = A^{(1,4)}b$$ I take the advantage of ...
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How to find the values of transformstion and its center/axis? [duplicate]

I have system of two planes. Each plane is defined by three points, so I have their equations. One of these plane is stable, I can't perform any transformation. The second one is modifiable - rotation ...
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1answer
125 views

Does having a zero eigenvalue preclude a matrix from being indefinite?

If a $3\times3$ matrix has a positive eigenvalue, a negative eigenvalue, and a zero eigenvalue, is it then, by definition, indefinite? I think so, since the matrix has both a positive and a negative ...
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386 views

A Tricky Math Question

Along a road lie an odd number of stones placed at intervals of 10 metres.These stones have to be assembled around the middle stone. A person can carry only one stone at a time. A man carried the job ...
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178 views

Solve the system using Gaussian elimination with back-substitution or Gauss-Jordan elimination

Here is the system: $$ \left\{ \begin{aligned} x_1-3x_3&=-2 \\ 3x_1+x_2-2x_3&=5 \\ 2x_1+2x_2+x_3&=4 \end{aligned} \right. $$ This is my very first problem actually using a matrix so here ...
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49 views

How many zero elements are there in the inverse of the $n\times n$ matrix

How many zero elements are there in the inverse of the $n\times n$ matrix $A=\begin{bmatrix} 1&1&1&1&\cdots&1\\ 1&2&2&2&\cdots&2\\ ...
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1answer
27 views

Linear Algebra Linear transformation Help

If $T:\mathbb{R}^n \rightarrow \mathbb{R}^n$ is a linear transformation, then there exists a basis for $\mathbb{R}^n$ in which $T$ is diagonal. Is this true or false
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36 views

Addition to make V = { $0, 1,\cdots,2^n-1$} a vector space over $GF(2)$

Let V = { $i \in \mathbb{Z} \mid 0 \leq i < 2^n$} for some positive integer n. How can you define vector addition and scalar multiplication to make this a vector space over $GF(2)$? This problem is ...
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1answer
33 views

When vectors go in a cycle pattern

If a combination of vectors, say $AB+DC+FG$ form a pattern that looks something like this: Does it mean that the outcome is the 0 vector? Bonus: Is there a way to denote this empty vector, much ...
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1k views

Proof of Gaussian elimination/Why does it work

I have just had a class on linear algebra and the professor explained how to solve matrixes. While he could explain how to solve them by using Gaussian's elimination, he failed to explain how does ...
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115 views

Percentage Contribution To Euclidean Distance

I am currently working with Euclidean Distances. I am calculating the distance between two n-dimensional sets of data points, but I really want to know how much each point contributes to the final ...
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3answers
74 views

Proving dot product of two vectors can differ

I am asked to prove when if vectors $u \cdot v = u \cdot w$ then it must follow that $v = w$ If I cannot prove it, I am asked to bring a counter example. If $u \cdot v = u \cdot w$ then that implies ...
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3answers
57 views

Let $Q$ be a $m \times n$ matrix. Prove $\det (Q Q^T) = 0$ if $m > n$.

Reading a book on another mathematical subject a proof makes use of linear algebra. Without explanation the author states the following: Let $Q$ be a $m \times n$ matrix. If $m > n$ then $\det ...
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30 views

Projection of a set described by half-spaces (linear inequalites)?

Let $C$ be a set of linear inequalities in $x-y$ plane, which defines a region on the space. Two questions: 1) How to check if $C$ is satisfiable? 2) If $C$ is satisfiable, how to compute ...
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1answer
41 views

How to derive the Descartes equation of a line in a coordinate system? $ax+by+c=0$

How to derive the Descartes equation of a line in a coordinate system? $ax+by+c=0$ I searched in proofwiki but I didn't find, and in KhanAcademy and youTube but I didn't find anything related to the ...
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1answer
49 views

Matrices - is this matrix in reduced row echelon form?

I have a matrix: $$\begin{bmatrix}1&0&0\\0&0&0\\0&0&1\end{bmatrix}$$ Im just wondering if it is or not...
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1answer
190 views

Solving an equation involving a trace: find $X$ in $M=\mathrm{Tr}[CX]$

So, I have this algebra problem: I have an equation of the form $$M_{ij}=\sum_{A,B}C_{ij}^{AB} X^{AB} \equiv \mathrm{Tr}\big[C_{ij}X\big]$$ where upper upper-case letters are some kind of ...
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Why does the Fourier series of $x$ not seem to give the right value?

I'm reading a lecture about Fourier series , and it says that you can represent any continuous function as Fourier series. There's a given example: Let $f(x) = x$. $f(x) \approx ...
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2answers
53 views

Are those rings fields?

Let $f(x) = x^4+x^2+1 \in Z_{2}$ and $A = Z_{2}/f(x)$ Is $A$ a field? Let $f(x) = x^4-x^2+1 \in Z_{7}$ and $B = Z_{7}/f(x)$ Is $B$ a field? I know that a ring $A = Z_{n}/f(x)$ is a field ...
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1answer
165 views

Linear combination of matrices

Let $A, B, C, D$ be four linearly independent symmetric 3 x 3-matrices over $\mathbb K$. Show that some linear combination of these matrices is a matrix of rank 1. I know it is supposed to be a ...
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56 views

Combinatorial exercise

A group of 15 people go visit a city with 150 bar. At the end of the day one of those bar contains 8 people, the another one contains the other 7 people. How many ways can we get this situation? ...
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92 views

Invariant Factors of the Zero Linear Transformation

Show that the zero linear transformation has invariant factors (and elementary divisors):$$q_1=x,q_2=x,\cdots,q_n=x$$ Here is my idea so far. If we have the zero linear transformation defined on ...
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192 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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1answer
115 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
136 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
291 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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171 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
43 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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1k 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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2answers
31 views

I need some help regarding definite matrix

Is a non-negative definite matrix a positive definite matrix?
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1answer
162 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
35 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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594 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
454 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
249 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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232 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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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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79 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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118 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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0answers
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
64 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 ...
4
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1answer
75 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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85 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