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 that the norm of $f'(y)$ is attained at $\pm\frac{\nabla f(y)}{\|\nabla f(y)\|}$.

Consider a $C^1$ function $f:\mathbb{R}^n\to\mathbb{R}$ and a point $y\in \mathbb{R}^n$ such that $\nabla f(y)\neq 0$. Prove that there exists an unit vector $x_0\in\mathbb{R}^n$ such that ...
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Conic reduction

I'm trying to reduce this conic : $x^2+y^2+2xy+x+y=0$ to a canonical form. I started with finding the eigenvalues of the matrix associated to the quadratic form $x^2+y^2+2xy$ I found $z_1=2 , ...
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11 views

Linear map and skalar multiplication

Let assume linear map $L: \mathbb{C}^3 \rightarrow \mathbb{C}^3$, defined as $Lu:=\langle v,u\rangle v$ where $u\in \mathbb{C}^3$ and $v\in \mathbb{C}^3$ is non-zero chosen vector with its norm ...
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18 views

Use separation of variables to find a solution $u= u(x,t)$…

So I get up to the last paragraph of the solution. I can get the bases of the solution, but beyond that, I'm really confused as to what they did. Any help would be appreciated!
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Field extensions and algebraic elements

Can somebody explain why taking beta gives $K(\beta)$ as a subspace of $K(\alpha)$?
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33 views

How do we find eigenvalues from given eigenvectors of a given matrix?

For instance let $$A=\begin{pmatrix} 3 & -1 & -1 \\ 2 & 1 &-2 \\ 0 & -1 & 2 \\ \end{pmatrix}$$ be a matrix and $$u_1=\begin{pmatrix} ...
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39 views

How do I show there exists a real matrix T such that

We have the following matrices : $$A=\begin{bmatrix}2 & 1 &-1\\1 & 2 & -1 \\-1&-1&4\end{bmatrix}$$ $$B=\begin{bmatrix}1 & 0 &-4\\0 & 5 & 4 \\-4&4&3 ...
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28 views

Mixture of mixtures.

I am formulating this problem for work, so it is important that I get it right. As of right now I am only considering the case where the number of chemicals is equal to the number of pre-made ...
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43 views

Is complete metric space is required?

This question may be quite related to the following link: but I am not sure. Sorry, if it is trivial. Advantage/disadvantage of complete/incomplete metric space. In many application specially in ...
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51 views

Computing eigenvalues and eigenvectors

I need a little help with this question: A is this self-adjoint matrix $$\pmatrix{ 1 & 0 & i \\ 0 & 2 & 0\\ −i & 0 & 1}$$ Compute the eigenvalues and a set of ...
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32 views

“positive matrices” in Sylvester's criterion

From Wikipedia: Sylvester's criterion states that a Hermitian matrix ''M'' is positive-definite if and only if all the following matrices have a positive determinant: the upper left 1-by-1 ...
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25 views

Let $A,B$ be $n\times n$ matrices. Assume $\det A=-2,$ $\det B\neq 0.$ $C=(A^{T})^{2}BA^{3}B^{-1}A^{-3}$. Calculate $\det C.$

Let $A,B$ be $n\times n$ matrices. Assume $\det A=-2,$ $\det B\neq 0.$ $C=(A^{T})^{2}BA^{3}B^{-1}A^{-3}$. Calculate $\det C.$ I hate to ask a question where I haven't at least made some headway, ...
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55 views

Canonical Mapping in Linear Algebra

I'm just really really wondering....is there such a thing? I mean, in my lecture notes, it's defined as, for a vector space $V$, and it's subspace or submodule $U$, to be the mapping $f:V \to V/U$, ...
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38 views

Vandermonde determinant and linearly independent

Let $a_1,a_2,a_3,b_1,b_2,b_3,b_4,b_5,b_6\in \mathbb{C}$ such that $a_i\not=a_j$ for all $i\not=j.$ If $$\begin{vmatrix} a_1 & a_2& a_3 & b_1 \\ a_1^2 & a_2^{2} & a_3^{2} & ...
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28 views

QR decomposition Q and R matrix

I had the vectors $(1,0,1),(-1,1,1),(1,0,1),(-1,1,1)$ and performed the orthogonalization process and got the orthonormal vectors: $$(\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}), (-\frac{1}{\sqrt{3}}, ...
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42 views

Let $V$ be a subspace of $R^3$ spanned by the vectors $(1,2,1)$ and $(2,1,2)$

a) Find an orthogonal basis for $V$. b) Find the projection of the vector $(1,3,0)$ onto $V$. c) Find the distance of the vector $(1,3,0)$ from $V$. Alright, i think i got it, but i guess what I'm ...
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30 views

Least Square method, find vector x that minimises $ ||Ax-b||_2^2$

Given Matrix A = | 1 0 1 | | 1 1 2 | | 0 -1 -1| and b = $[1\ \ 4\ -2]^T$ find x such that $||Ax - b||_2^2$ is minimised. I know I have to do something along the line $A^TAx = A^Tb$ got the ...
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49 views

How to project gradient vector to subspace defined by linear constraints

I have the following set of linear constraints: $$\begin {align}\textbf{y}^T\textbf {x} &= 0 \\ \textbf {0} &\leq\textbf {x} \leq C\cdot\textbf {1},\end {align}$$ where $\textbf {y} \in ...
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23 views

Intuition behind direction of maximum variance?

I'm trying to understand the phrase "direction of maximum variance" which keeps popping up in the context of PCA. For example, in this set of 2D points, it is clear they approximately lie on a line. ...
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40 views

Determine whether the set S (below) is a subspace of $M_2(\mathbb{R})$ (The space f all 2x2 matrices with real entities)

$$S = \left\{ \begin{pmatrix} 2a && -b \\ 3b && a \end{pmatrix}\;:\;\; a,b \in \Bbb R\right\}$$ So, I have to find that it is either closed under addition and scalar multiplication, I ...
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25 views

equation with double fractions

$$ \frac{\pi + \frac65\pi}{2}=\frac{11}{10}\pi $$ I've found this equation in some of the examples, we've got this equation, so as I think: $$ \frac{\frac{5}{5}\pi + ...
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Invariant subspaces of $\begin{pmatrix}1 & 1\\ -1 & 1 \end{pmatrix}$

How many invariant subspaces $$\begin{pmatrix}1 & 1\\ -1 & 1 \end{pmatrix}$$ has? There are at least 2 invariant subspaces: $\mathbb{R}^2$ and $\{0\}$. The matrix of the operator is ...
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Is there an endomorphism $F$ and a vector $v$, such that $F$-invariant $span\{v, F(v), F^2(v), …\}$ is decomposable

If $V'$ is any vector space, can a vector $v' \in V'$ and an endormorphism $F: V' \rightarrow V'$ exist, such that the $F$-invariant subspace $U = span\{v',F(v'), F^2(v'), ...\}$ is decomposable into ...
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25 views

$Im(A+B) \subset ImA + ImB$

Consider linear maps from $\mathbb{R}^n$ to $\mathbb{R}^m$, where $m,n \geq 2$. A. $Im(A+B) \subset ImA + ImB$ B. $\ker(A+B) \subset \ker A + \ker B$ I figured out a counterexample for ...
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39 views

Space of Tikhonov regularization of an Ill poised problems.

This question is motivated by the answer given in the following link: What space to use? My understanding from the above answer is that given a problem and the properties we want the solution to ...
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21 views

Existence of solution to a matrix inequality

Let $n\in\mathbb N$ and $\mathbf G\in\mathbb R^{n\times n}$ be a symmetric, positive definite, non-singular matrix. Let $\mathbf c\in\mathbb R^{n}$ be an arbitrary vector. Does there exist a vector ...
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42 views

What does norm of a matrix mean?

I was reading the proof of SVD decomposition form here SVD decomposition proof. I was able to follow the proof except for one thing, they define norm of a matrix as $$|A|_2= \text{sup}_{v_1 \in ...
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33 views

If $A$ is a $k \times k$ submatrix of $n \times n$ unitary matrix and $2k>n$. why that some singular value of $A$ is equal to $1$

If $A$ is a $k \times k$ submatrix of $n \times n$ unitary matrix and $2k>n$. why does some singular value of $A$ is equal to $1$?
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18 views

$\sum _{i=1}^n \sum _{j=1}^n x_i x_j a_{i,j}$ same as quadratic form

I have the quadratic forms equation: I was wondering if the summation would be equal to: $$\sum _{i=1}^n \sum _{j=1}^n x_i x_j a_{i,j}$$ Or is there a special reason to represent the sum in the ...
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49 views

Compute the determinant $D_n$

I would like to compute: ...
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59 views

Proof involving vectors spaces and endomorphisms

I'm not sure how to go about this problem. Let $U$ be a finite dimensional vector space over a field. There exists some $T \in End(U)$ such that $T^2 = T$. Show that $U = U_1 + U_0$ with $T(u_1) = ...
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25 views

When told “The system reduces to one in row echelon form” what does this mean?

Does this mean that in its reduced form it is an identity matrix? Or is it describing something different?
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Finding Linear Independence in $P_3$

We are asked to deteremine whether the vectors are linearly independent: $ x^2 + 1, x + 1, x^2 + x $ in $P_3$ I began the problem as follows: Let $ S $ = { $x^2 + 1, x + 1, x^2 + x$ } in $P_3$ ...
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15 views

Projection out of orthogonal matrices

Let A,B be orthogonal matrices of order $n \geq 2 $. $\det A = 1, \det B = -1$. There exist $a \in [0,1]$ such that $aA + (1-a)B$ is projection. I know that the claim above is false. I ...
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21 views

Complete eigenvalues

I need to confirm if the eigenvalue given is a complete eigenvalue and also need to determine the dimension of the associated eigenspace. I know that an eigenvalue is complete if the geometric and ...
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42 views

Doubt about a spanning set solution.

I have given a set $N = \{0, 1, x, 2x^2, 3x^3, 4x^4, 5x^5\}$. I know that this set is linearly dependent, because it contains the $0$ polynomial. That is why it can not be a basis for $R_5[x]$ ...
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33 views

Show that every real number lambda is an eigen value of L

Let S be the set of all sequences $$(x_1,x_2,x_3,...)$$ of real numbers. Consider the linear left-shift map: $$L:S \rightarrow S:L(x_1,x_2,x_3,...)=(x_2,x_3,x_4,...)$$ Show that every real number ...
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22 views

Finding the eigenvectors of a matrix that has one eigenvalue of multiplicity three

This is a simple question, which hopefully has a quick answer. I have a given matrix A, such that \begin{equation} A = \begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 1 \\ 0 & 0 & 1 ...
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40 views

Does $\text{End}_K(K^n) \cong \text{Mat}(n\times n, K)$?

Let $K$ be a field, $K^n$ a vector space over $K$. Is the following true? $\text{End}_K(K^n) \cong \text{Mat}(n\times n, K)$ Does this change if $K$ is a ring, and $K^n$ a module over $K$?
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What is meant by a “$S$-strictly diagonally dominant matrix” in the book 'Geršgorin and His Circles'

What is meant by a "$S$-strictly diagonally dominant matrix" in the book Geršgorin and His Circles. The definition of strictly diagonally dominant is easy to find, but the definition of ...
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Diagonalisation and direct sums

Let $U$ be a finite dimensional vector space. Let $T:U\to U$ be a linear transformation with eigenvalues $\lambda_1,...,\lambda_n$. Then $T$ is diagonalisable if and only if ...
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56 views

Bound for eigenvalues of some special matrix

Let $Tridiagonal(a, c, b)= \begin{vmatrix} c & b & 0 & \ldots & 0 \\ a & c & b & \ldots & 0 \\ 0 & a & c & \ldots & 0 \\ \vdots & \vdots & ...
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34 views

When is the matrix of eigenvectors of a complex symmetric matrix orthogonal?

Given a complex symmetric matrix $A=A^\top$ with a matrix of eigenvectors $C$ (which have distinct non zero eigenvalues) it can be shown that $C^\top C=I$ and that $C^\top A C=D$ where $D$ is a ...
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16 views

How many isometries in R3 that…

In the euclidian field $R^3$, how many isometries apply $(1,0,0)$ on $(\sqrt{2}/2,\sqrt{2}/2,0)$ and $(0,1,0)$ on $(0,0,1)$ ? I am tempted to answer only one, the one which apply $(0,0,1)$ on ...
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32 views

Decompose an invertible $4 \times 4$ real matrix into product of $4 \times 3$ and $3 \times 4$

If we have an invertible matrix $M$ that is $4 \times 4$ and $\left| M \right| \neq 0$ (i.e. it is invertible), is it possible to decompose it into two matrices $4 \times 3$ and $3 \times 4$ ...
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44 views

What all possible matrix reprsentations a linear operator can have?

$L$ is a linear operator such that $L:V \to V$ where $V$ is a $n$ dimensional hilbert space. If $[L]_{ij}$ is the matrix representation for $L$ in the input and output basis $\{i\}$ and $\{j\}$, then ...
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19 views

Subspace, Direct Sum, Polynomials, Basis

Let $U = \{p \in \mathcal{P}_4(\mathbb{F}) \;\colon\; p''(6) = 0\}$. a. Find a basis for $U$. b. Extend the basis in part (a) to a basis for $\mathcal{P}_4(\mathbb{F})$. c. Find a subspace ...
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22 views

Birth-death process: What is the distribution of reached states before reaching an absorbing state?

Intro I am working on a birth-death process. For a given choice of parameter ($n=6$, $Wa=1$, $Wb=0.95$, see below), the transition matrix is $$\left( \begin{array}{ccccccc} 1. & 0.144928 & ...
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42 views

Determine whether or not T is a linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^4$

Let $T : \mathbb{R}^3 \to \mathbb{R}^4,$ with $T([x, y, z]) = [x + y + z, z − y, x − y, x + 1].$ I want to determine whether or not $T$ is a linear transformation. I've tried to figure out how to ...
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26 views

Angles between points in $3$D space where the Origin is not the vertex.

Given two points $P_1,P_2$ in $3$D space that are positioned around a third point $M$, how do you calculate the angle between $P_1,M,P_2$. I know there are a few questions on here discussing how ...