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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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
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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
184 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
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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
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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
14 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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17 views

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
104 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 ...
4
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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
34 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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2answers
14 views

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
34 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 ...
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1answer
30 views

Isomorphism matrix problem

So the question asks: Recall that $U^{2\times 2}$ is the vector space of 2X2 upper triangular matrices. Which of the following functions are isomorphisms? A. The function T: $U^{2\times 2}$ to ...
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1answer
25 views

How polynomials are represented in matrix form for Univariate Polynomial. [on hold]

Represent this polynomial equation in matrix form $$P(x)=a_2 x^{2} +a_1x^{1} +a_0$$ ?
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Linear Algebra - Changing bases?

Let $T: \mathbb{R}^3\rightarrow M_{2\times2}(\mathbb{R})$ be the linear transformation defined by $T((a,b,c)) = \begin{bmatrix}a&5a\\c&3c\end{bmatrix}$. Consider the bases ...
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3answers
28 views

Explanation on derivation of this equation?

I'm extremely stuck on how my book was able to derive this equation. Basically, it says: Let $V = W = P_2(\mathbb{R})$. A basis for V is $1, 1+x, 1+x+x^2$. Define the linear transformation $T$ such ...
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1answer
23 views

Eigenvalues of matrix summation

Let $A$ be symmetric positive definite matrix with eigenvalues $\lambda_1,\lambda_2,\dots,\lambda_n$. Can we express the eigenvalues of $I-A$ using eigenvalues of $A$? I can't find properties of ...
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How to show that a set of elements is a basis for the ring of integers of a number field?

Let $K$ be a number field of degree $n$ (that is $[K:\mathbb{Q}]=n$) with ring of integers $\mathcal{O}_K$. I know that there exists algorithms to find $\mathcal{O}_K$ and hence determine a ...
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3answers
45 views

Spectrum Radius

Let $A$ be symmetric positive definite matrix and $\rho(A)$ denotes spectrum radius of $A$. How to prove that $\rho(I-\omega A)<1$ iff $0<\omega<2/\rho(A)$?
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1answer
33 views

Why are these functions linearly dependent?

$f1(x) = x$ $f2(x) = x^2$ $f3(x) = 5x - 4x^2$ From my understanding a set of functions are only linearly dependent if you can show that one function is simply a scaled version of another in the ...
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0answers
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Linear Algebra Solving for Matrices

Consider the following matrix. $A$ = [ a b c ] [ d e f ] [ g h i ] Suppose that $\det(A) = −2$. Let $B$ be another $3 \times 3$ matrix (not ...
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2answers
14 views

Condition on matrix to ensure nontrivial Jordan canonical form

In my understanding, in order to ensure that a matrix $A$ has a nontrivial Jordan canonical form, one needs to come up with such a matrix whose geometric multiplicity is less than algebraic ...
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1answer
20 views

Proving Linear Independence Given Odd Absolute Values

With three vectors $a,b,c \in \mathbb{R}^3$, the magnitude of a$,b,c,a-b,b-c$, and $c-a$ are all odd integers (not necessarily distinct). How could you prove the three vectors are linearly ...
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Is this how to do matrix representation?

Say, $$f: \mathbb{Q}[t]_{4} \to \mathbb{Q}[t]_{4}$$ $f(q)=3q'''+2q''$ And we have the base $B=\{1,t,t^2,t^3,t^4\}$ and we wanted to find $[f]_{B}^{B}$ Then is this what would we do; $$f(1)=0$$ ...
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1answer
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Stable eigenspace of $x'=Ax$

Given the system $\bf{x'}=Ax$, where $\bf{A}$$=\begin{bmatrix} -2 &0 &0 \\ 2& 1 & 0\\ 0 &0 &1 \end{bmatrix}$, the solution is $x(t) = \begin{bmatrix} e^{-2t} & 0 ...
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condition for having a positive solution to these linear equations.

Consider the following system of linear equations: $\displaystyle\sum_{j=1}^n c_{ij}\cdot x_{ij}=a_i$ for $j=1,\cdots,m$ and $\displaystyle\sum_{i=1}^m c_{ij}\cdot x_{ij}=b_j$ for $j=1,\cdots, n$ ...
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$\bf{x'}=Ax$ with eigenvalues of multiplicity greater than $1$

Given the system $\bf{x'}=Ax$, where $\bf{A}$$=\begin{bmatrix} -2 &0 &0 \\ 2& 1 & 0\\ 0 &0 &1 \end{bmatrix}$, if I solve it by first finding matrix $\bf{P}$ and then ...
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3answers
63 views

Compute a natural number $n\geq 2$ s.t. $p\mid n \Longrightarrow p^2\nmid n$ AND $p-1\mid n \Longleftrightarrow p\mid n$ for all prime divisor p of n.

Question: Compute a natural number $n\geq 2$ that satisfies: For each prime divisor $p$ of $n$, $p^2$ does not divide $n$. For each prime number $p$, $p-1$ divides $n$ if and only if $p$ divides ...
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2answers
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Basis of a Subspace given an Equation.

Hi, I was wondering if this question is asking us to find the basis of the kernel of this transformation from $\mathbb{R}^4 \rightarrow \mathbb{R}$ ? Thanks
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1answer
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A linearly independent set that spans a space

So, in partial differential equations, we generate solutions for PDEs (kind of obviously). However, while the solutions we generate span the space of all solutions and are all linearly independent, ...
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How can I rearrange this formula to give it in terms of $t$? [on hold]

How can I rearrange the equation $$ e^{2t} = \frac{y^{2}(y+1)}{y-1} $$ to give it in the form $y = f(t)$?
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3answers
62 views

Linear maps (about rank and nullity)

$S:U\rightarrow V$ and $T:V\rightarrow W$ are linear maps. $U,V$ and $W$ are vector space over the same field. Prove: If $V=W$ and $T$ is non-singular then ...
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1answer
24 views

Distance between projections

Let $x,y,z \in \mathbb R^2$ such that $||x|| = ||y||= ||z|| = 1$. Project $z$ onto the lines defined by $x$ and $y$ as follows: \begin{equation} z_x = (z^\text{T}x) x, \ \ z_y = (z^\text{T}y) y, ...
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1answer
10 views

Subspace of $C^3$ that spanned by a set over C and over R

Given $A=$ $\left\{ {(1,2 + i,i),(1,3 + i,3 - i),(i,3i,4 + i)} \right\}$ Let $SP_CA$ be the linear space spanned by A over $C$ Let $SP_RA$ be the linear space spanned by A over $R$ what is the ...
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37 views

Is it possible to find determinant of a matrix by given the eigenvectors and the eigenvalues

If I already found the eigenvalues and eigenvectors of a particular matrix , is there an easy way to find the determinant of that matrix ?
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What is $\left | \left | A \right | \right |$ equals to in linear algebra?

Can someone please tell me what is this $\left | \left | A \right | \right |$ equals to? (determinant inside determinant)
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Proving that union of epsilon nets is an epsilon net

While reading a paper, I came across these definitions and claims: Definition: Given $p \in \mathbb R^d$, and $H$, a set of hyperplanes, let $$\text{Violate}_p(H) = \{h \in H: h \text{ is strictly ...
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Bounding the perturbation between eigenvectors

Can somebody explain this part of the proof of a deduction from the Davis-Kahan $\sin \theta$ theorem? I understand how to get from: $||P_{u_1} - P_{v_1}|| \le \epsilon$ to $||P_{u_1}v_1 - v_1|| \le ...
2
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1answer
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Linear Algebra - properties of positive semidefinite matrix

Let $A=(a_{ij})$ be a positive semi-definite, symmetric matrix, of order $3\times 3$ satisfying: $$ \Sigma_{j=1}^{3} a_{ij}=0 $$ for $i=1,2,3$ (i.e.- the sum of each row is zero). Prove: ...
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0answers
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Bounding entries of random vector

Given a random vector $\mathbf{e} \in \mathbb{R}^n$, is it possible to count (or bound) the number of entries in $\mathbf{e}$ that have $|e_i| \ge 1/ \sqrt{n}$? It is known that entries in ...
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1answer
36 views

Linear Algebra. Is this question realte to combination and factorials?

I am not able to understand this question and what is the entries of matrix A exactly. Question Thanks.
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52 views

Linear vs. bilinear

I'm tripping over something elementary: Suppose $f:\mathbb{R^2}\rightarrow X$ is linear, then $f(x+y)=f(x)+f(y)$ for all vectors $x$ and $y$. Now suppose that $f$ is also bilinear and in particular ...
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1answer
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$A$ is a linear transformation with eigenvalues

Let $A:\mathbb{R}^2\rightarrow\mathbb{R}^2$ be a linear transformation with eigenvalues $\frac{2}{3}$ amd $\frac{9}{5}$. Then, there exists a non zero vector $v\in\mathbb{R}^2$ such that ...
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$f: L \to L$ a diagonalizable operator with simple spectrum and $g: L \to L$ s.t $gf = fg$

I am making the exercises of Kostrikin and Manin (Linear Algebra and Geometry) and it has this question that I can't solve. Let $f: L \to L$ a diagonalizable operator with simple spectrum and $g: L ...
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1answer
21 views

Show linearity of this map, if and only if statement

Let $Γ_f = \{(s, t)\mid t = f(s)\} \subset S \times T$ Suppose that $U$ and $V$ are vector spaces and $f ∈ V^U$ (i.e. f is a map of underlying sets which is not necessarily linear). how do you show ...