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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Proving adjugate of $A$ for $3 \times 3$

From Wikipedia's article on adjugate matrix, Cayley–Hamilton theorem allows the adjugate of $A$ to be represented in terms of traces and powers of $A$. For the $3 \times 3$ case: ...
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3 views

Find the homothetic transformation

In $\mathbb{R^3}$: Find the homothety $\Phi$, such that the following transformations are possible: $$\Phi(P)=\Phi(1,0,-1)= (2,5,0)$$ and $$\Phi(Q)=\Phi(0,1,2)= (0,5,2)$$
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1answer
10 views

Matrix rank and number of linearly independent rows

I wanted to check if I understand this correctly, or maybe it can be explained in a simpler way: why is matrix rank equal to the number of linearly independent rows? The simplest proof I can come up ...
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2answers
24 views

Find $\det(A)$ of Matrix and condition on a and b

Let $$ A=\begin{bmatrix} a & b & 1 \\ b & 1 & b \\ 1 & a & a \\ \end{bmatrix} $$ Find $\det(A)$ in terms of $a$ and $b$, and write down ...
3
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3answers
19 views

$A$ has more columns than rows and has full row rank, show there exist infinitely many $B$ s.t. $AB=I$

If A $\in M_{m\times n}(R)$ such that $n>m$. Prove that if $\text{rank} (A) = m$ then there are infinitely many matrices $B \in \ M_{n\times m} (R)$ such that $ AB = I_m$ So the question is ...
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15 views

Is a metric's form determined by its signature?

Suppose that we define a 4-dimensional vector space over the real field with a metric with signature (3, 1). Is the scalar product map determined only with this information? For example: a Minowsky ...
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47 views

Prove that $A+2I$ is invertible [duplicate]

Given $A$ is a square matrix such that $A^{3} = 2I$ Prove that $A+2I$ is invertible and find its inverse. How do I prove that $A+2I$ is invertible? For proving $A-I$ is invertible, I use ...
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24 views

Simpler way to show $v$ must be zero?

Let $x$ and $y$ be linearly independent vectors in $\mathbf{R}^2$. If $v \in \mathbf{R}^2$ is orthogonal to both $x$ and $y$, then $v$ is the zero vector. Here's my proof: Since $x$ and $y$ are ...
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3 views

Stability criterion for eigenvalues of an AR(2) process.

This is pretty much a question on linear algebra stemming from time series analysis. Essentially I want to find a stationarity criterion for an AR(2) process. It is easy to reduce this to the ...
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16 views

Solve $KA-BK=0$, for a $1 \times n$ dimension row vector $K$, where $A$ is known $n \times n$ matrix and $b$ is known scalar

The above equation with mentioned dimensions is to be solved. How can I find the value (or approximate value) of row vector $K$. Please help.
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Receiving different answers

Ok, so im following a tutorial on how to calculate a limit numerically and when the tutor plug'd in the number $(-1.1)$ into the equation $\frac{(t^6 -1)} {(t^3 + 1)}$ HE gets −2.331 as the ...
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1answer
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$S_1 \subset S_2$. To show, $Span(S_1) \subset Span(S_2)$

Prove that if $S_{1} \subset S_{2}$, then $Span(S_{1}) \subset Span(S_{2})$ Approach: Suppose $S_{1} \subset S_{2}$ Let $x \in S_{1}$, then by definition of a subset, $x \in S_{2}$ All possible ...
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16 views

simultaneous diagonalize

$A = \begin{pmatrix} 18 & -9 \\ -9 & 9 \end{pmatrix}$ $B = \begin{pmatrix} 3 & 2 \\ 2 & -2 \end{pmatrix}$ Find a real invertible matrix such as $P^tAP = I_2$ and $P^tBP$ is diagonal ...
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How to find the inverse of the Haar (4) matrix? [on hold]

$$ H_4 =\begin{bmatrix} 1 & 1 & 1 & 1\\ 1 & 1 & -1 & -1\\ 1 & -1 & 0 & 0\\ 0 & 0 & 1 & -1 \end{bmatrix} $$
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3answers
73 views

Is $\rho(A^2) = \rho(A)^2$?

How can I show that $\rho(A^2) = \rho(A)^2$? Is that even true? I´ve tested it with matlab for random matrices, and the equation was always true. I´m pretty sure that even $\rho(A^n) = \rho(A)^n$ ...
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1answer
17 views

Picture of vector in $R^3$ and vector in $R^2$ reflected across plane [on hold]

I have trouble imagining what reflecting a vector in $R^2$ and a vector in $R^3$ across x-y plane and y-z plane look like. Would you please draw me a picture?
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1answer
32 views

Does $A-\lambda I$ have rank smaller than $A$?

Consider $\lambda$ as one eigenvalue of $A$, can we say that $A-\lambda I$ must have rank smaller than $A$? Or equivalently, $A-\lambda I$ spans a space which is a subset of $A$?
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1answer
13 views

Linear independence and Wronskian - Proof or Counterexample

If $y_1(x) , y_2(x) ,\ldots,y_n(x)$ are linearly independent in $C[b,c]$ then they are Linearly Independent in $C[a,d]$, where $a<b<c<d.$ So I know if the Wronskian isn't zero for at least ...
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5 views

Gradient Descent: L2 Norm Regularization

So I've worked out Stochastic Gradient Descent to be the following formula approximately for Logistic Regression to be: $ w_{t+1} = w_t - \eta((\sigma({w_t}^Tx_i) - y_t)x_t) $ $p(\mathbf{y} = 1 | ...
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Find only the real eigenvalues of a matrix.

If a matrix has many (thousands) complex and few (dozen) real eigenvalues is there a fast method for estimating only the real eigenvalues ?
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1answer
25 views

Find the vector form of the equation of the line in $\mathbb{R}^2$ that passes through $P = (2, -1)$

Find the vector form of the equation of the line in $\mathbb{R}^2$ that passes through $P = (2, - 1 )$ and is parallel to the line with general equation $2x - 3y = 1$. Following the format of $x = p ...
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30 views

Verify rotation relation between two matrices

Suppose we have two matrices how do we verify that one of them is related to the other by a rotation, $$AU = B$$ where $UU^T=I$. One way is to form $AA^T$, and $BB^T$ and see if they are equal. How ...
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77 views

Determinant of this matrix? [on hold]

How can I find the determinant of this matrix?
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23 views

The sum of two subspaces

Let $V_{1}$ and $V_{2}$ be two subspaces of V. Define the sum of $V_{1}$ and $V_{2}$ to be the subset of V $V_{1}+V_{2}=${$\overrightarrow v_{1} + \overrightarrow v_{2}:\overrightarrow v_{1} \in ...
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1answer
9 views

Magnitude of orthogonal projection

I have a basic linear algebra question. Suppose that $ u \in \mathbb{R}^n $, and $ P(u, V) $ is the orthogonal projection of $ u $ onto a linear subspace $ V $. I would like to prove that $$ ...
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28 views

System with parameters [on hold]

How can I solve the following system where $\lambda$ and $\mu$ are parameters and what is the answer?
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17 views

Find (linear) transformation matrix using the fact that the diagonals of a parallelogram bisect each other.

This is the first time I post something on this website. I'm on this question already for hours. I'm clearly not asking the community to do my homework, I'm hoping someone can explain me how I should ...
4
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1answer
43 views

Number of invertible matrices modulo 26

The number of invertible matrices modulo $26$ can be computed by the Chinese Remainder Theorem. i.e. a matrix is invertible modulo 26 if it is invertible modulo $13$ and modulo $2$ which are given ...
4
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1answer
60 views

Finding the Determinant of a particular Matrix

I've come across the question of finding the determinant of the $(n\times n)$ matrix, given by $$A:= \begin{pmatrix} x & 1 & 1 & \dots & 1 \\ 1 & x & 1 & \dots & 1 \\ ...
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Is $\text{Rank }(T) = \text{Dim}(V)$ all the time?

Thm Let $V$ and $W$ be Vector spaces and let $T:V \to W$ be linear If $\beta = \{ v_1,\dots ,v_n \}$ is a basis for $V$ then $$ R(T)=\text{span}(T(\beta))=\text{span}(\{ T(v_1),\dots,T(v_n) \} ...
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Solution of $x'=Ax$ is not what it is supposed to be

Consider a system of ODEs $x'=\begin{bmatrix} 0 &1 \\ -1&0 \end{bmatrix}x$. Wolfram Alpha says that the solution is $x(t) = \begin{bmatrix} \cos t &\sin t \\ -\sin t& \cos t ...
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3answers
25 views

Understanding Defiinition of Vector Space

Let $F$ be a field. A vector space over $F$ is a set $V$ together with $+$,$\cdot$ satisfiyng: $$+: V \times V \rightarrow V$$ $$\cdot: F \times V \rightarrow V$$ with usual properties. My ...
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Minimizing the error by finding optimum step-size

I need to recheck a proof for minimizing the error by finding optimum step-size. I re-checked the proof many times but still can't find a mistake although the number I am getting in Matlab is not ...
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11 views

Perturbation theory for a symetric rank-one update

I know perturbation theory of the eigenspectrum/singular value decompostion of a symetric matrix $A$ under a symetric perturbation $E$, that besides being symetric has no other structure. Is there ...
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$ax=0$ if and only if $a=0$ or $x=0$ [duplicate]

Prove that $ax=0$ $\Leftrightarrow$ $a=0$ $\lor$ $x=0$, where $a$ is a scalar from a field and $x$ is an element of the vector space on this field. I would like a hint or maybe a solution to prove ...
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1answer
24 views

Linear algebra MOOCs

I am a statistics student studying a module of linear algebra at the undergrad level. I was looking for MOOCs that might help me. I tried saylor which meets my syllabus but I cannot find videos for ...
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4answers
167 views

Let A be a square matrix such that $A^3 = 2I$

Let $A$ be a square matrix such that $A^3 = 2I$ i) Prove that $A - I$ is invertible and find its inverse ii) Prove that $A + 2I$ is invertible and find its inverse iii) Using (i) and (ii) or ...
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2answers
19 views

Collection of linear combinations of linearly independent vectors

If we have linearly independent vectors $v_1, v_2, ..., v_n$ and create a new collection of vectors $v_1', v_2',...,v_n'$ such that each $v_i'$ is a linear combination of $v_1, v_2, ..., v_n$. Are ...
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0answers
15 views

Condition for nullity of quadrilinear form

I have been told the following. Lemma Suppose $V$ is a vector space over a field $K$, and $T:V\times V\times V\times V\to K$ is a multilinear map with the following properties holding for all ...
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1answer
12 views

Is it true that solving a triangular system using forward or backward substitution numerically stable?

The system is $TX = B$, where $T$ is a triangular matrix, $X$ is a unknown matrix, and $B$ is the RHS matrix. I know the system $Tx = b$ is backward stable where $b$ is a RHS vector. Detail check ...
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Function from $\mathbb{R}^9$ to $\mathbb{R}^6$ with zero set the orthogonal $3\times 3$ matrices

I am trying to construct a $C^\infty$ function from $\mathbb{R}^9$ to $\mathbb{R}^6$ with zero set the orthogonal $3\times 3$ matrices. I am thinking about mapping $M$ to $MM^T-I$, but am not sure ...
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1answer
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Calculating 5 different ranges for people resource management

I am working on a project for my company. My team is building a project charter template. In this template needs to be a drop down that estimates how many full-time employee days(FTE) will be ...
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XOR binary matrix multiplication $AX=B$? [on hold]

Let $A$, $B$, and $X$ be binary matrices (in F2 ), where $A$ and $B$ are of size $n \times m$ with $n > m $. $X$ is an $m \times m$ matrix. Compute $X$ such that $AX=B$. ps: $A$ is not a ...
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Matrix of rotation

I am haunted by a question. Consider a vector $v=\begin{bmatrix} a\\ b \\ c \end{bmatrix}$ is firstly multiplied by $R_1= \begin{bmatrix} \cos(\theta_1) &-\sin(\theta_1)&0\\ \sin(\theta_1) ...
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4answers
101 views

Let $S = \{n\in\mathbb{N}\mid 133 \text{ divides } 3^n + 1\}$. Find three elements of S.

Question: Let $S = \{n\in\mathbb{N}\mid 133 \;\text{divides} \; 3^n + 1\}$ $a)$ Find three different elements of $S$. $b)$ Prove that $S$ is an infinite set. My intuition is find the prime factors of ...
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1answer
39 views

Proving dimension formula in linear algebra

Let $V$ and $W$ be finite dimensional vector spaces and let $T:V \to W$ be a linear transformation. (a) Prove that if $\dim(V) < \dim(W)$ then $T$ cannot be onto. (b) Prove that if ...
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1answer
32 views

Graded Vector Spaces (definition)

I am studying Algebraic Operads with the book Algebraic Operads, by Jean-Louis Loday and Bruno Vallette and I'm having a little problem with the definition of graded vector space. My advisor and I ...
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Prove a covariance matrix is positive semidefinite

Given a random vector c with zero mean, the covariance matrix $\Sigma = E[cc^T]$. The following steps were given to prove that it is positive semidefinite. $u^T\Sigma u = u^TE[cc^T]u = E[u^Tcc^Tu] = ...
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Realation between Matrices ?!

The answer to the following question could be trivial. Let $A_1, A_2$ be symmetric $n\times n$ matrices, $x=(x_1,\ldots,x_n)\in \mathbb{R}^n$. If the maximum is taken for over ($\|x\|=1,\, and ...
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2answers
33 views

Solution Set of System of ODE.

I am trying to find the solution of the system $$\begin{bmatrix}x_1\\x_2\end{bmatrix}'= \begin{bmatrix}1&3\\3&1\end{bmatrix}\begin{bmatrix}x_1\\x_2\end{bmatrix}$$. I am given that ...