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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How to prove the following

Let $\mathbf{A}\in\mathbb{R}^{p\times n} (n\ge p)$ be a positive definite symmetric matrix having a Wishart distribution with mean $\mathbf{0}$ and covariance $\boldsymbol\Sigma\otimes \mathbf{I}$. ...
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Proving the existence of an inverse of a matrix. (Linear algebra)

Suppose that $A$ has no inverse. Prove that there exists a vector $b$ such that $Ax = b$ has no solution My try Proving by contradiction , Assume that for all vector $b$, $Ax = b$ have at least one ...
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Proof for linearity on tensor products

Theorem: Let $U$ and $V$ be vector spaces. Let $\mathbf{u}^* \in U^*$. Define $\mathbf{f} : U \otimes V \to V$: $$\mathbf{f}\left(\sum_{r} \mathbf{u}_r \otimes \mathbf{v}_r\right) = \sum_{r} ...
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$|f(x)−f(y)|≤u(a,b)|x−y|^t$ for all $a≤x,y≤b$. For what values of $t$ is $V_t$ a subspace of $\mathbb{R}^\mathbb{R}$?

For any real number $0<t≤1$, let $V_t$ be the set of all functions $f∈\mathbb{R}^\mathbb{R}$ satisfying the condition that if $a<b$ in $\mathbb{R} $then there exists a real number $u(a,b)$ ...
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1answer
10 views

Choleski decomposition of a positive matrix

Let us consider a matrix $\boldsymbol{F}$. We consider its Choleski decomposition, $ \boldsymbol{F} = \boldsymbol{M} \boldsymbol{M}^T $. We know that $\boldsymbol{F}$ needs to be positive definite. ...
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Symplectic matrix

I want to show that if $\lambda$ is a real eigenvalue of a symplectic matrix $A$ then its char poly is of the form $det(A-\mu id) = (\lambda-\mu)(\frac{1}{\lambda}-\mu) det(\hat{A}- \mu id) $ where ...
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Determining the intersection of kernel and image.

I was posed the following question: If $T$ is a linear operator on a finite dimensional vector space $V$ such that rank of $T$ = rank of $T^2$. I'm supposed to show that the kernel and image of $T$ ...
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I need to write equations for 2 rates of change. [on hold]

here is my problem, Write the following as an equation. x / y 1 / 3 2 / 12 3 / 27 4 / 36 5 / 51
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13 views

Finding a variable in the determinant of sum of matrices

I Don't Know how to earn P that is scalar from Below Formula : R = log2(abs(det(I + P * H*H'))) Everything is known except P. P is scalar and positive. I is an Identity NxN , H is complex NxN ...
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$(1)$ $a\boxplus b=a+b-ab$ for all $a,b \in F$. $(2)$ $a\boxdot b=1-t^{\log_t{(1-a)}\log_t{(1-b)}}$ for all $a,b\in F$

Let $1<t\in \mathbb{R}$ and let $F=\{a\in \mathbb{R}: a<1\}$. Define $\boxplus$ and $\boxdot$ on $F$ as follows: $a\boxplus b=a+b-ab$ for all $a,b \in F$. $a\boxdot ...
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Measuring the effect of a linear transformation on the result

I have an unknown vector $x\in\mathbb{R}^n$, a known orthogonal matrix $\Phi\in\mathbb{R}^{n\times n}$, a known matrix $A\in \mathbb{R}^{m\times n} (m \le n)$, and a known vector $b\in \mathbb{R}^m$ ...
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1answer
9 views

eigenvalues of cycle graph and its complement graph

I am trying to find the eigenvalue of cycle graph and its complement. How to simplify.Suppose $\omega^{1}+\omega^{n-1}=2cos (2\pi/n) $, then, $\omega^{\frac{n-1}{2}}+\omega^{\frac{n+1}{2}}=?$ Is it ...
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18 views

Need help about field decision; mathematics or physics? Who can be good at these? [on hold]

First of all, you may want to delete this question because it is not an mathematical question, but this question can be an opening door to thousands of mathematical question. Hello everybody, I need ...
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1answer
15 views

Finding jordan normal form

Let be $T:\mathbb{R}^7\rightarrow \mathbb{R}^7$ Such that $(T-15I)^3=0$ and $\dim\text{Im}(T-15I)^2=2$ find the Jordan normal form of $T$ If $(T-15I)^3=0$ so the minimal polynomial can be ...
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1answer
36 views

Is $\left \{ x_{1}+y_{1},…, x_{n}+y_{n}\right \}$ a basis for $\mathbb{R}^{n}$?

Suppose $\left \{ x_{1},..., x_{n}\right \}$ and $\left \{ y_{1},..., y_{n}\right \}$ are two different bases for $\mathbb{R}^{n}$. Is $\left \{ x_{1}+y_{1},..., x_{n}+y_{n}\right \}$ also a basis for ...
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3answers
24 views

Linear Independence and Subset Relations

I've been reading the wikibook on Linear Algebra and in the section 'Linear Independence and Subset Relations' it defines the following lemma: Lemma 1.14: Any subset of a linearly independent ...
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1answer
22 views

An orthogonal matrix has eigenvalue $1$ with the eigenspace $E(1)$ of dimension $n-1$. Then $-1$ is also an eigenvalue with $E(-1)$ of dimension $1$?

Let $(V,g)$ be an $n$-dimensional Euclidean space ($g$ scalar product) and let $f:V \to V$ such that $g(f(u), f(v)) = g(u,v)$. It is known that the matrix associated to $f$ with respect to an ...
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1answer
18 views

Show that a set of polynomials are linearly independent in the complex space

I have been trying the solve the following question without any success: Let $\lambda_1, \lambda_2, \lambda_3$ be three distinct complex numbers and define the polynomials $m(\lambda), m_1(\lambda), ...
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1answer
12 views

Question about forced damped oscillators

This question is from my linear algebra book: Find the function $f(t)$ of the form $f(t)=a \cdot cos(2t)+b \cdot sin(2t)$ such that $f''(t)+2f'(t)+3f(t)=17cos(2t)$ All I've figured out so far is ...
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1answer
22 views

Notation and name for this function?

Let $k \geq 1$; let $V,W$ be vector spaces; and let $T: V \to W$ be linear. Then how do we call and denote the function $(v_{1},\cdots, v_{k}) \mapsto (T(v_{1}), \cdots, T(v_{k})): V^{k} \to W^{k}$?
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Looking for the basis of the kernel of T

Let P$_2$ denote the vector space of all polynomials with real coefficients and of degree at most 2. Define a function T : $P_2$ → $P_2$ by $$ T(P(x)) = x^2 \frac{d^2}{dx^2}(p(x-1))+ ...
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28 views

How do i show V is a linear subspace if it's defined like this?

Let {$V =\{(x,y,z) \in \mathbb R^3 : x+3y=3z\}$,and let $T :V \to \mathbb R^3$. be given by $T(x,y,z)=(x,y,z)\times(1,3,−3)$, the usual cross-product in $\mathbb R^3$. How do i show that V is a ...
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1answer
26 views

For any linear operator $\phi$ on $V$, prove such an integer $m$ exists.

Suppose $V$ is an $n$-dimensional vector space over some infinite number field $K$, $\phi\in\mathcal L(V)$, prove there exists such a (positive) integer $m$ that $$\text{Im} \phi^m=\text{Im} ...
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1answer
42 views

Show that $Ax=0, Bx=0$ share the same solution space iff there is some invertible $P$ s.t. $B=PA$.

The question is said in the title, suppose $A,B\in M_{m\times n}(K)$, where $K$ is some infinite number field. If we regard $A,B$ as linear maps from $K^n$ to $K^m$, then they share the same ...
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Solving a system of ODEs with 4 repeated eigenvalues

I'm working on problem which requires me to solve a system of ODEs with 7 equations. I've gotten as far as determining the eigenvalues and vectors of my coefficient matrix $A$, but 4 of the ...
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1answer
20 views

How do you show that f(z)=z conjugate isn't linear?

let $x_1= a+ib,x_2= c+id,k=$scalar $f(x_1,x_2)=f(x_1) + f(x_2)$ $f(a+ib + c + id)=(a+c)-i(b+d)$ $f(a+ib)+f(c+id)=(a+c) - i(b+d)$ $f(kx_1)=kf(x_1)$ $f(k(a+ib))= k(a-ib)$ $kf(x_1)=k(a-ib)$ Looks ...
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1answer
7 views

Minimum of restricted linear combinations.

Let $\{N_0, ... , N_m\}$ be a set of natural numbers, then the minimum $(\geq 1)$ of all their linear combinations is their GCD. Is there a way to calculate that minimum if some $N$s can only be ...
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Given the tetrahedron $OABC$, find a condition on $a OA+ b OB + c OC$ such that this is always inside $ABC$.

I did the following: Taking the tetrahedron $OABC$, one can decompose it in: $OA,OB,OC, AB,BC$. And then, writing: $$x(BC-AB)+AB\quad x\in[0,1]$$ We obtain all the points in the line segment from ...
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1answer
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I want to appeal this problem from an exam in Linear Algebra I, do you think its appealable? [on hold]

I have the follow question : Let $U_1, U_2, W$ are linear spans of linear space $V$ while V is finite. Proof: If $$U_2 \cap W \neq \{0\}$$ $$U_1\cap W\neq \{0\}$$ $$U_1 \cap U_2=\{0\}$$ Then $dimW ...
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1answer
27 views

Complex conjugate of $z$ as a different variable

Can a complex conjugate be represented by a different letter than $z$? As in: Let $y$ be a complex number satisfying $|y|<1$. Find the set of all complex numbers $z$ satisfying ...
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1answer
31 views

What is a change of basis and how do i find it?

W is a four dimensional vector space over a field F with basis S = (v1, v2, v3, v4). B is also a basis of W such that. $b1 =−v1, b2 =v1 +v2, \, b3 =−v1 −v2 −v3, \, and \, b4 =v1 +v2 +v3 −v_4.$ ...
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1answer
27 views

Where do the variables of a quadratic form live?

Consider a quadratic form $Q = a_{ij}x_{i}x_{j}$, where the summation from 1 to $n$, the number of independent variables is implied on $i$ and $j$. By this definition $a_{ij}$ is not symmetric, but ...
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Is the following optimization problem a way to find SVD? [on hold]

Let $B$ a matrix of size $n \times m$. I want to show that the SVD, top $k$ singular vectors, can be found by solving: $$\max_{\displaystyle{\begin{array}{c}||u_i|| = 1, i \in \{ 1,\ldots, k \},\\ ...
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A question about product of three positive definite matrices

Assume that $A,B$ and $C$ are symmetric positive definite matrices. I guess that the eigenvalues of the matrix $D=ABC$ can be any complex numbers. Is that true?
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17 views

which of the following is an equivalence relation of the set S

which of the following is an equivalence relation of the set S I have solved all except d and need your help please
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1answer
27 views

Matrix polynomials/eigenvalues

$\begin{pmatrix} 7 & -2\\2 & 2 \end{pmatrix}$ The eigenvalues for this matrix are $\lambda=6$ and $\lambda=3$ It also happens that $(A-6I)(A-3I)=0$ I've checked for various $2$ x $2$ ...
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Jordan normal form over $\mathbb{C}$

Let there be $T:\mathbb{C}^8\rightarrow \mathbb{C}^8$ Such that $ T\left(\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \\ x_{5} \\ x_{6} \\ x_{7} \\ x_{8} ...
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Disprove Officer's account - Velocity / Distance / Time [on hold]

I am looking for a mathematical proof which definitively invalidates a false account of events, re: fail to stop - red light. If anyone is interested, thank you kindly, and please let me know if I ...
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Exercise of algebra [on hold]

Given the linear transformation $T:\mathbb{R}^4 \to P_2(\mathbb{R})$ such that: $\ker(T) = \{(x_1,x_1,x_3,x_4) \in \mathbb{R}^4 : 2x_2 - x_3 + x_4 = 0,\, 2x_1 - x_2 = 0\}$ $T(0,1,0,1) = -2x^2 + x$ ...
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A question on matrix norm

Definitions: ${A_j},{\Delta _j} \in {C^{n \times n}},(j = 0,1,2....m)$ ${\rm{P(}}\lambda {\rm{) = }}{{\rm{A}}_m}{\lambda ^m} + .....{A_1}\lambda + {A_0}$ is a matrix polynomial, and $\lambda $ is ...
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1answer
42 views

Three lines that intersect in a plane.

Find a condition for three lines (𝑖 = 1,2,3) in a plane given by $𝑎_𝑖 𝑥 + 𝑏_𝑖 𝑦 = 𝑐_𝑖$ to intersect in one point. I decided to form a matrix and to find the identity matrix since it will ...
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1answer
45 views

Exercise of algebra II [on hold]

Can anybody please help me with this exercise?My exam is comming soon :S It says: 1)We define = f: P₂[R] ---> R^2x2 linear transformation whose transformation matrix in basis B and E' is: Mf(over ...
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2answers
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if n is a positive integer let Z be the subset of integer in {1,…,n} which are relatively prime to n

if n is a positive integer let Z be the subset of integer in {1,...,n} which are relatively prime to n my effort to solve this question I''m confused and need help to solve this question please ...
2
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1answer
37 views

How does $u^Tv = p \cdot \|u\|$ follow from the projection onto line?

Before anybody asks, this is not a homework question. I just saw the formula given in Andrew Ng's Coursera course in the SVM section. For reference: the projection formula is $$ \mathrm{proj}_w(p) = ...
5
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1answer
19 views

How to show the sum of the images of such $m$ projections is direct and is the whole space?

There are $m$ projections (whose square are themselves) $\phi_1,\cdots,\phi_m$ acting on a finite-dimensional vector space $V$ such that $$\phi_i\phi_j=0\quad i\ne j\tag{1}$$ where $0$ denotes the ...
3
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2answers
51 views

Does the set of all piecewise constant functions form a subspace of the vector space $\mathbb{R}^\mathbb{R}$ over $\mathbb{R}$?

A function $f\in \mathbb{R}^\mathbb{R}$ is piecewise constant if and only if it is a constant function $x\to c$ or there exist $a_1<a_2<\cdots<a_n$ and $c_0,...,c_n$ in $\mathbb{R}$ such that ...
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0answers
37 views

Solving for a 3D point in a 5D graph given 3 pairs of 2D points.

I am attempting to solve the values $C$, $D$, and $S$, given three pairs of $[M,R]$. $$R = \frac {M}{C - MDC + DC\left(MS\right)^2}$$ I have been able to solve for a related equation (or rather, ...
2
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3answers
26 views

Linear independence for a set of real valued continuous functions

Let $V$ be the vector space of all real valued continuous functions. Is the following set $\{\cos t, \sin t, \mathrm{e}^t\}$ linearly independent? I usually understand what and how to determine ...
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0answers
24 views

Moving object position

An object is moving with 20 points p/s. Currently the object is at position x: 30, y: 50, z: 90. The object is moving to x: 4^6, y: 4^8 z: 9. What are the coordinates after 25 minutes? This is what ...