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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Derivates and Limits in the Same Problem are an Issue.

I am working on the following problem:- Evaluate lim x→1 [( x^1/4 - 1 ) / ( x^1/3 - 1 )] by relating it to the derivatives of functions. Now this is quite a ...
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1answer
10 views

Show that if $(QA)x = 0$ has just the trivial solution, then $A$ is invertible

Let $Q$ be an invertible $n\times n$ matrix, $A$ an $n\times n$ matrix and $x$ is an $n\times 1$ column vector (or matrix) so that the matrix equation $Ax = 0$ represents a homogeneous system of $n$ ...
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1answer
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If we know nullspace of matrix, how to find reduced row echelon form of that matrix?

vectors u = [4 1 0 0] and v = [1 0 2 1] form a base of nullspace of matrix $$ A\in M_{5,4}(R) $$ Find a reduced row echelon form of Matrix A. Since $ n-r = dimN(A) $ we know we got two base ...
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Matrix Algebra $(A+B)^2$ help

Does $(A + B)^2$ = $A^2$ + $B^2$ or $A^2$ + $B^2$ + $AB$ + $BA$ ? Where A and B are both matrices.
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1answer
8 views

Quadratic forms — rank of matrix

Assume that $M$ is the matrix of some quadratic form (over any field of characteristic not $2$) and set $$Q(\overline{x})=\overline{x}^tM\overline{x}$$ We can replace $M$ by the symmetric matrix ...
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2answers
126 views
+50

Maximum square cells in a rectangle

I know this sounds like bin packing but it's a bit different so please read the question to the end. Given a rectangle of known width and height, I need to divide it into smaller rectangles using ...
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1answer
31 views

for $k\neq 0, -1, 1$, find the inverse of the matrix

for $k\neq 0, -1, 1$, find the inverse of the matrix $$\begin{pmatrix} k&0&0\\ 1&k&1\\ -1&1&k \end{pmatrix}$$ how am I supposed to solve this? all I can think of is plugging ...
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6answers
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In plain language, what's the significance of a field?

I just started Linear Algebra. Yesterday, I read about the ten properties of fields. As far as I can tell a field is a mathematical system that we can use to do common arithmetic. Is that correct?
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21 views

Find whether vector w belongs in the span

$$v_1=[1,0,1,2]$$ $$v_2 = [0,1,1,3]$$ $$v_3 = [2,1,3,7]$$ $$w = [1,2,3,4]$$ We are supposed to determine if $w$ is in $\operatorname{span}(v_1,v_2,v_3)$.
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2answers
30 views

The Number of Increasing Vectors $(x_1,…,x_k)$ Satisfying $1 \leq x_i \leq n$ and $x_1 < x_2 <…<x_k$

I want to find the number of increasing vectors $(x_1,...,x_k)$ satisfying $1 \leq x_i \leq n$ and $x_1 < x_2 <...<x_k$. Examples of vectors satisfying these conditions Let $n =5$ and ...
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Basis of $\mathbf{Q}[x]$ [duplicate]

I wanna show that the binomials $\binom{x}{k}$ for $k=0,1,\ldots$ form a basis of the $\mathbf{Q}$ vector space $V=\mathbf{Q}[x]$. I can show that for fixed $m\in\mathbf{N}$ the $\binom{x}{k}$ ...
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8 views

Study the associative and commutative properties and neutral and inverse elements of these groups

Group m*n = max(m,n) on Z and N So i showed its associative by m,n,p in Z and (m*n)*p = max(m,n)p =max(m,n,p) And m(n*p) = m*max(n,p) = max(m,n,p) Commutative m*n = max(m,n) and n*m = max(n,m). I ...
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0answers
16 views

What is the identity matrix in N dimensional space?

For example I have a matrix A of shape [2,3,3]. What would be the matrix I, such as: A*I = A [2,3,3] means A is a 3-dimensional matrix where each dimension has 2, 3 and 3 cardinal. Here is the ...
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1answer
16 views

Deduction of vector form of Snell's law

I was unable to find the deduction of the vector form of Snells's law. $$n_1\sin\theta_1 = n_2\sin\theta_2$$ Here is the vector form, from the article A Theory of Multi-Layer Flat Refractve Geometry ...
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0answers
8 views

Use of mathematica to put equation into vector form

Is there a way to put the following equation of a line into vector form using mathematica (or like mathematical package)? $\displaystyle ...
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0answers
11 views

Evaluate this covariance matrix.

Let $X$, $Y$ be two multivariate random variables with mean values $\mu_X,\mu_Y\in\Bbb{R}^d$ respectively. We assume the above mean values are given as follows $$ \mu_X=\Big( ...
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1answer
25 views

$AB*\text{adjoint}(BA)=I$

$AB*\text{adj}(BA)=I$ Prove: $1$. $|AB|=1$ $2$. $AB=BA$ As for $2$. what I have menage is $AB*AB^{-1}=AB^{-1}*AB=AB*\text{adj}$(BA)=I$ \rightarrow BA=AB$ How do I solve $1$. and is $2$. is ...
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2answers
44 views

Is it true: A real symmetric matrix is either positive definite or negative definite or indefinite?

I got a real symmetric matrix that is neither positive definite or negative definite, so can I just say that this matrix is indefinite? Thanks in advance:)
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8 views

Is T well-defined? Find the matrix representation.

Let $W_1$ be the subspace of C(0,1) spanned by the functions $\{e^x,xe^x,x^2e^x\}$. Let $W_2$ be the subspace of C(0,1) spanned by the functions $\{1,e^x,xe^x,x^2e^x\}$. Let T be the application ...
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1answer
52 views

General formula for $\det(A+I)$ where I is identity. Worked it out for $2 \times 2$ and $3 \times 3$.

Does anybody know a general formula for $|A+I|$ where $A$ is a $\textbf{symmetric}$, real (square) matrix? For a $2\times2$ system I worked out: $|A+I| = |A|+\text{tr}(A)+1$. This is very friendly. ...
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4answers
20 views

Showing linear dependence in a matrix

Let $v_1=\begin{bmatrix} 1\\ -3\\ 2\\ \end{bmatrix} $ Let $V_2=\begin{bmatrix} -3\\ 9\\ -6\\ \end{bmatrix} $ Let $V_3=\begin{bmatrix} 5\\ -7\\ h\\ \end{bmatrix} $ For what value of h is the span ...
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21 views

What is the proof/show that the post of linear transformation generated by LDA is at most k-1

What is the proof/show that the matrix $Sw$ generated by LDA is at most rank $p-k$, where $p$ is the dimension of the data and $k$ is the number of classes. LDA: ...
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0answers
7 views

Prove if B is semidefinite then B11 is semidefinite

If B is a p x p symmetric matrix = (B11 B12 B21 B22 ) and B is semidefinite. Prove B11 is semidefinite where B11 is a q x q matrix
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16 views

Complex conjugate of a unitary matrix

Is there any example for which complex conjugate of a unitary matrix is not a unitary matrix ?
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1answer
117 views

Proof of Laplace expansion using minors

I've come across with the following proof of the Laplace expansion: Let $\Delta=\sum_{j=1}^n (-1)^{1+j} a_{1j}\bar M_j^1$ and $\tilde{\Delta}= \sum_{j=1}^n (-1)^{i+j} a_{ij}\bar ...
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3answers
45 views

Proofs about Matrix Rank

I'm trying to prove the following two statements. I can prove them easily by considering the matrix as a representation of a linear map with a given basis, but I don't know a proof which uses just the ...
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0answers
6 views

Is it possible to scale a matrix by some amount to make the semi-major axis of the error ellipse equal to the semi-major axis of the unscaled matrix

I am working on a matrix problem that involves calculating the error ellipse from a covariance matrix. Suppose I have some covariance matrix in the Earth-Centered Earth-Fixed (ECEF) reference frame, ...
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2answers
56 views

Is the set of invertible upper triangular matrices open in $GL_n(\mathbb R)$? Is it open in the set of all upper triangular matrices?

I think the answer to the second question is yes, but can't quite prove this. I've no idea about the first part. I've done a few exercises of this kind but all have used the continuity of the ...
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2answers
24 views

Description of real projective spaces in various contexts

What I want to know is : What is the description of real projective spaces (specially $RP^0$, $RP^1$, $RP^2$) respectively in context of topology, geometry and algebra? I'm searching for simple ...
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0answers
23 views

linear algebra question

Consider $n$ convex polytopes $S_1, \cdots, S_n$ and a set of matrices $W$ such that each matrix $A\in W$, we have that the $i$-th row of $A$ is a member of $S_i$. (In general $W$ is infinite.) ...
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2answers
30 views

Is it true that $\biggl\|I-\frac{vv^T}{v^Tv}\biggr\|=1$?

I am following a proof in the text OPTIMIZATION THEORY AND METHODS a springer series by WENYU SUN and YA-XIANG YUAN. I come across what seems obvious that for a column vector $v$, with dimension ...
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1answer
22 views

jordan canonical form of 6x6 matrix

Let Nl and N2 be 6 X 6 nilpotent matrices over the field F. Suppose that N1 and N2 have the same minimal polynomial and the same nullity. Prove that N1 and N2 are similar. Show that this is not true ...
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34 views

Syndrome Leaders

I've been stuck on this problem for a long time and I can't figure it out. I need to find the set of coset leaders and their syndromes. I have my coset leaders (I think) but I can't figure out how to ...
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1answer
29 views

Searching for analytical or topological proof(s) of the Cayley-Hamilton theorem

Is there any analytical or topological proof(s) of the Cayley-Hamilton theorem ? I want to know such proofs ( if possible ) , I would even appreciate proper references with accessible links . Thanks ...
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1answer
17 views

How to prove that every negative value is an eigenvalue for T?

Let $V$ be the linear space of all functions continuous on $(-\infty,\infty)$ and such that the integral $\int_{-\infty}^x tf(t) \,dt$ exists for all real $x$. If $f\in V$ let $g=T(f)$ be defined by ...
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0answers
15 views

Is a complex function really just an infinite dimensional matrix?

I have recently sort of come to the understanding that integrating two functions multiplied together is a sort of infinite dimensional dot product, and I only know this from taking an undergraduate ...
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3answers
27 views

Short question about the homogenous system

I am working on a text book problem for a intro linear course. But the solution is not in the back. I am looking to see if I understand it correctly. The question asks, " If A is a matrix, and the ...
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1answer
38 views

Matrices over PID

Let $R$ be a PID and $A,B\in\operatorname{M}_n(R)$ are $n\times n$ matrices such that $\det(A)\sim\det(B)\neq0$,i.e., the ideals generated by $\det(A)$ and $\det(B)$ are the same, does there exist ...
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1answer
222 views

If $(V,k)$ is a finite-dimensional vector space, then the space of all linear transformations on $V$ is finite dim and find its dim?

My issue with this is the only way I know how to prove it is to set $\dim V=n$, but then that wouldn't make sense because the second part is find the $\dim$. What I was thinking is using the ...
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1answer
42 views

Finding maximum no of $1$'s

We are given a matrix $A \in M_n (F)$ such that all its entires are either $1$ or $0$. I need to find the maximum number of $1$'s that can be in matrix $A$ so that it is still invertible. My try : ...
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1answer
258 views

The rank of QR factorization

If A is a $m\times n$ matrix $m\geq n$,A=QR is a reduced QR factorization. If R has k nonzero diagonal entries ($0\leq k<n$). I want to know what is the rank of A.Is it at least k?
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Is there a binary [10,6,4] code?

Using the sphere padding bound formula I can conclude that 1 + 12 + 66 $\ge$ $2^{6}$ which indicates that there MAY be a binary [10,6,4] code, however I cannot prove that there is. How can I come to ...
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1answer
34 views

$dim_\mathbb C V=n$ then $dim _\mathbb R V=2n$

Prove that if the dimension of a vector space $V$ over $\mathbb C$ is $n$ then the dimension of $V$ over $R$ is $2n$ I wanted to do it using isomorphisms i.e. every finite dimensional vector space ...
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1answer
18 views

Invariant probability vector as a left eigenvector

What is the probability in the long run that the chain is in state 1? Solve this by directly computing the invariant probability vector as a left eigenvector. \begin{bmatrix} .4 &.2 &.4 \\ ...
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15 views

Finding a parity check matrix of a binary code

I'm supposed to find a parity check matrix of a binary [6,3,3] code. Given a generator matrix G i can find a parity check matrix by row reducing until I get the identity matrix, then take -A^{\top} | ...
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1answer
42 views

Determinant in $\mathbb Z_{5}$

I need to find $$ \det\left[ \begin{array}{cc} 2 & 4 & 0 \\ 1 & 1 & 3 \\ 3 & 2 & 1 \end{array} \right] $$ over $\mathbb Z_{5}$ What I did: $$2\det\left[ ...
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Distance, Speed and Time problem [on hold]

A tourist was traveling from town A to town C but decided to take a detour she could pass through town B. Forty minutes after she left town A, she noted that the remaining distance to town B was twice ...
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1answer
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Find the distance [on hold]

The US presidential convoy stretches 4miles and is moving in to Boston. A special protective vehicle at the back was given a letter to deliver at the front of the convoy. After delivering the letter ...
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1answer
21 views

Finding a standard generator matrix given a binary code

My question is how do I find the standard generator matrix of a binary [7,6,2] code? From what I understand a generator matrix for $C$ is any $ k \times n$ matrix $ G$ with entries in $ ...
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35 views

What does it mean that the product of two vectors produces real number?

I am going over inner product space. I know that linear space has an inner product as long as it satisfies $4$ conditions. And, the book says that for $x,y$ in $V$, there is a real number ...