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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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, ...
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1answer
6 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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2answers
79 views

How to solve this system of nonlinear equations?

How to solve these equations for $a$, $b$, $c$ and $x$? I have the following: \begin{align} 1 &= 2a+b+c\\ a &= (a+b)x + 0.25(a+c)\\ a&=(a+c)(1-x)\\ b&=a(1-x)+c(x-0.25)\\ ...
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1answer
15 views

Determining which vectors are solutions of a given system of equations.

Determine which vectors are solutions of the system. \begin{align*} & \hphantom{+}3x-2y-5z = \hphantom{+}4 \\ & \hphantom{+}2x+4y-\hphantom{1}z = \hphantom{+\llap{$0$}}2 \\ & {-}4x-8y+9z ...
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1answer
13 views

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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3answers
22 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
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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0answers
15 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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0answers
8 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
20 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
32 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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1answer
29 views

Legendre transform is everywhere finite iff $ f$ grows faster than $ 2$-norm

Let $f:\mathbb{R}^n \to \mathbb{R}\cup \{ \infty \}$ be convex. Its Legendre transform is $f^* (d):=\sup_{x\in \mathbb{R}^n}(d^Tx-f(x))$ Show $f^*(d)<\infty$ $\forall d\in \mathbb{R}^n$ iff ...
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1answer
9 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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31 views

Show that for every $n\times n$ matrix $A$, there exist an $n\times n$ permutation matrix $P$ such that $PA$ has $LU$ factorization

I am stucked at this problem: Prove by induction on $n$ that for every $n\times n$ matrix $A$, there exist an $n\times n$ permutation matrix $P$ (a matrix obtained by rearranging the rows (or ...
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1answer
21 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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1answer
23 views

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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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
25 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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compare norms on $\mathcal{B}(H)$

Given a Hilbert space $H$ and $a$ be a real numbers $\geq‎‎‎ 1$ , let $S_1(H)$ denote the space of trace-class operators on $H$, with the trace-class norm or Schatten 1-norm. That is $$ \Vert T ...
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4answers
903 views

Why is the matrix product of 2 orthogonal matrices also an orthogonal matrix?

I've seen the statement "The matrix product of two orthogonal matrices is another orthogonal matrix. " on Wolfram's website but haven't seen any proof online as to why this is true. By orthogonal ...
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1answer
26 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
49 views

Prove that any $n \times n$ matrix can be written as in block form

Prove any $n \times n$ matrix can be written as in block form: $\begin{pmatrix} N & 0 \\ 0 & B \end{pmatrix}$ where $N$ is a $k \times k$ nilpotent matrix ($N^n=0$) and $B$ is an ...
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3answers
88 views

If $\{x_i\}_{i=1}^n$ are the roots of $f(x)=a_nx^n + a_{n-1}x^{n-1} + \ldots +a_0$ then $\sum_{i=1}^nx_i^{n-1}$ is independent of $a_0$

I found an interesting conclusion when I did this simple question. Let $$f(x)=(x^2-1)(x+2)=x^3+2x^2-x-2$$ and let $x_i$ for $i=1,2,3$ be the roots of $f(x)$. Find the sum $\sum\limits_{i=1}^3x_i^2$. ...
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1answer
19 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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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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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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0answers
8 views

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
39 views

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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0answers
27 views

$|f(x)-f(y)|\leq u(a,b)|x-y|^t$ for all $a\leq x,y \leq b$. For what values of $t$ is $V_t$ a subspace of $\mathbb{R}^\mathbb{R}$?

For any real number $0<t\leq 1$, let $V_t$ be the set of all functions $f\in \mathbb{R}^\mathbb{R}$ satisfying the condition that if $a<b$ in $\mathbb{R}$ then there exists a real number ...
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1answer
24 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
26 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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0answers
16 views

Is the following optimization problem a way to find SVD?

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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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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2answers
96 views
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A question about an infinite sequence of elementary row operations

Do there exist matrices $A$ and $B$ such that $B$ can be transformed into $A$ only if an infinite number of elementary row operations are performed on $B$? "What can we multiply the top equation by ...
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0answers
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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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1answer
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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2answers
47 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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2answers
300 views

Random directions on hemisphere oriented by an arbitrary vector

Hy, i'm writing a raytracer, and for that I need to generate n random vectors that are inside an hemisphere oriented by the surface normal. Ideally, I would also like being able to restrict the rays ...
2
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0answers
22 views

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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1answer
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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 ...
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) = ...
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0answers
24 views

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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0answers
42 views

Exercise of algebra [on hold]

Given the linear transformation $T:\mathbb{R}^4 \to P_2(\mathbb{R})$ such that: $\ker(T) = \{(x_1,x_1,x_3,x_4) \in \mathbb{R}^4 : 2x_2 - x_3 + x_4 = 0,\, 2x_1 - x_2 = 0\}$ $T(0,1,0,1) = -2x^2 + x$ ...
3
votes
1answer
335 views

Vector Project onto Subspace

So the question is: Let S be the subspace of $\mathbb{R}^3$ spanned by the vectors $ u_2 = \begin{pmatrix} \frac{2}{3}\\\frac{2}{3}\\\frac{1}{3}\end{pmatrix} u_3 = \begin{pmatrix} ...
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1answer
36 views

Let $V=\{i\in \mathbb{Z}: 0\leq i< 2^n\}$. Define vector addition and scalar multiplication on $V$ to turn it into a vector space over $GF(2)$.

Let $V=\{i\in \mathbb{Z}: 0\leq i< 2^n\}$ for some $n\in \mathbb{N}$. Define vector addition and scalar multiplication on $V$ in such a way as to turn it into a vector space over the field ...
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2answers
44 views
+50

Linear mapping matrix with paramters.

I solved a linear mapping problem recently and it turns out no to be correct, although i thought it was a simple problem. The problem asks me to find real parameters $a,b,c$ such that linear mapping ...
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0answers
17 views

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 ...
-5
votes
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 ...