Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

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1answer
29 views

$\dim V = \dim \phi(V)+\dim \ker \phi$

I want to show that $\dim V = \dim \phi(V)+\dim \ker \phi$. I know this proof can be found in any linear algebra textbook. However, my question is not exactly about the proof, but on a statement I ...
0
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1answer
25 views

Transformation matrix between 2 bases

Given a matrix $A = \begin{bmatrix}1 & 1 & 1\\ 0 & 1 & 1\\ 0 & 0 & 1 \end{bmatrix}$ and bases to a the vector space $V$: $B=(v_{1},v_{2},v_{3}),\qquad ...
1
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0answers
10 views

Least squares with matrix in GF(2)?

Here's an example of a problem I'm working on involving finding combination of bit vectors that yield a certain sum (in the GF(2) sense): $ \begin{pmatrix} 1 & 1 & 1 & 1 & 0 & 0 ...
3
votes
5answers
47 views

Proof involving subspaces

I encountered this question in a document I found on a google search, it bugged me because my perception keeps telling me I'm wrong no matter what I do. Let $U$, $W$ and $Z$ be subspaces of a ...
2
votes
2answers
33 views

Efficient way of checking linear independence

Suppose I have a $4 \times 4$ matrix $A$ whose columns represent vectors $v_1,v_2,v_3,v_4$ in $\mathbb{R}^4$. Now, given that $\det{A} = 0$ (i.e. the vectors are linearly dependent), I want to make ...
1
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4answers
34 views

Quotient spaces in linear algebra

I'm having a bit of difficulty understanding what a quotient space is to a vector space $V$. I will present the part I'm finding trouble with below. Let $V$ be a vector space and let $U$ be a sub ...
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0answers
17 views

The equivalence of homogenous systems of linear equations in two unknowns that have the same solutions

I am self-studying Linear Algebra by Hoffman & Kunze. Exercise 6 in Section 1.2: "Prove that if two homogenous systems of linear equations in two unknowns have the same solutions, then they are ...
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0answers
22 views

Solving eqn. of the form K = AGL + BGT, where A,B,L,T are invertible matrices.

I am obtaining the following equation in a regression problem: \begin{eqnarray} Z'_1Y_1\Omega^{-1}_{1}A+Z'_2Y_2\Omega^{-1}_{2}A = Z_{1}'Z_1\Pi A'\Omega^{-1}_1A + Z_{2}'Z_2\Pi A'\Omega^{-1}_2A ...
1
vote
2answers
31 views

Find $x$ as the given $n$th term in the Fibonacci sequence?

With a given $n$ and I am trying to find the value of $x$, as in: $$Fib(x)=n$$ Using the formula for Fibonacci sequence, where $\varphi$ is the Golden Ration ($\approx1.61803399\ldots$) $$Fib(z) = ...
2
votes
1answer
20 views

Prove that $A$ is similar to $B$ probably using Jordan form

Let $A, B \in M_n(\mathbb{F})$ such that: $a_{ij} = 0 \iff b_{ij} = 0$. $a_{ij} = b_{ij}$ for all $i \ne j+1$ $\exists \lambda \in \mathbb{F}$ such that $a_{ii} = b_{ii} = \lambda$. Prove that ...
1
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1answer
14 views

showing a basis is suitable for a dual space, from linear forms

Let us define $B = b_1, b_2, b_3$; where $ b_1(f) = f(0)$, $ b_2(f)=−f'(0) $ and $b_3(f) = f''(0) $. Let $E ^∗$ be the dual basis of $E = {1, x, x^2}$. Show that B is a basis of the dual ...
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0answers
20 views

Vector for arcs in path

I have path created from lines and arcs. I want to create next path inside or outside of this given path with given offset. For line I calculate line equation and it gives me simple perpendicular ...
-1
votes
1answer
31 views

Solving three linear equations in terms of unknown

$$\alpha+\beta+\gamma=a$$ $$\alpha+\beta=b$$ $$\gamma=c$$ Find the values of $\alpha,\beta,\gamma$ in terms of $a,b,c$ Obviously, the value of $\gamma$ is $c$ So after eliminating $\gamma$ from ...
1
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2answers
28 views

Finding inverse linear transformation

I'm solving a homework question and I'm stuck with it's last part. The question goes like this: Let $\displaystyle T:M_n(\mathbb{R})\to M_n(\mathbb{R})$ be a transformation defined as ...
0
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0answers
17 views

How can we split a single rotation into two along orthogonal axes?

I have the following axis system, where the X-Y plane is horizontal and Z points 'up': I have a horizontal plane that I want to rotate so that the angle between it and the XY plane is theta. I ...
0
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0answers
27 views

Generalization of N-Body Problem

I know the n-body problem has been solved for gravity, but in a purely mathematical sense, has it been solved? Or could it be generalized to any kind of field? Maybe an example will make my question ...
1
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1answer
31 views

How to continue on proving that rank (A+B) ≤ Rank A + Rank B? [duplicate]

Theorem: rank (A+B) ≤ Rank (A) + Rank (B) Proof: Let U = Im(A)& W = Im(B). By dimension theorem, we know that: Dim(U+W) = Dim(U) + Dim(W) - Dim (U ∩ W). By substituting U and W we get: ...
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3answers
56 views

Prove that if $\mathbf A$ is an $n\times m$ matrix, then $\text{tr}(\mathbf A \mathbf A')=\text{tr}(\mathbf A' \mathbf A) $

If $\mathbf A$ is an $n\times m$ matrix, then $\text{tr}(\mathbf A \mathbf A')=\text{tr}(\mathbf A' \mathbf A) \text{ where } \mathbf A'\text{ is transpose of }\mathbf A\text{ and tr}(\mathbf A ...
0
votes
1answer
10 views

Proving that and LC of solutions is still a solution

I am currently using Lay's Lineair algebra and its functions, on page 316. On this page, I have the following problem. One page earlier is stated that a multiplication x' = Ax (where A is a matrix ...
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0answers
35 views

Prove that $A$ is similar to $B$

Let $A, B \in M_n(\mathbb{F})$ such that $m_A(x) = m_B(x)$ and $f_A(x)=f_B(x)=(x-\lambda_1)^{d_1}\cdots (x-\lambda_k)^{d_k}$ for different $\lambda_1, \ldots, \lambda_k$ such that $1 \le d_l \le 3$ ...
1
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2answers
30 views

Changing order of summation - proof

How was the right side of equation obtained from its left side? I could obviously guess immediately that this is true, but mathematics is not about guessing. Are there any intermediate steps between ...
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0answers
18 views

Find an orthonormal basis for the subspace w

Find an orthonormal basis for the subspace $ W = \text{span} \{(3, 0, 4, 0),(0, −2, 1, 0),(0, −3, 0, 1)\}$ of $\mathbb{R}^4$ Without using Gram-Schmidt process.
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1answer
58 views

How the picture of DETERMINANTS come up? [on hold]

Matrices represent some sort of linear transformation. If we consider a linear transformation from a space to itself they are called endomorphisms. I also read that determinants are used to measure ...
0
votes
1answer
35 views

At most $n+1$ vectors, the angle between which $>\pi/2$.

In a $n$ dimensional Euclidean space $V$, there exists at most $n+1$ vectors, each pair has inner product $<0$. This is geometrically obvious in $3$ dimensions...But how can we prove it ...
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0answers
31 views

Prove that $V_i$ are $T$-invariant for $1\le i\le k$ and $V=\bigoplus_{i=1}^{k}V_i$

Let $T$ be a linear operator over $V$ with dim$(V)=n$ and let the ordered set $B=${$v_1,v_2,...v_n$} be a basis for $V$. Furthermore, let $A=[T]_B$ be block-diagonal matrix (that is ...
0
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0answers
16 views

Improvement of Minimum description length (MDL) estimate.

I earnestly request apology if this question is inappropriate for the forum. The question has two parts one technical and the other is not technical. I would appreciate any response. Let me consider ...
1
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1answer
65 views

Steinitz's Lemma - Removing

In the book that I am using, Linear Algebra Done Right, the proof for the Steinitz exchange lemma (which can be found here) left me unconvinced. The proof refers to the linear independence lemma. ...
0
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0answers
7 views

How to calculate a covariance matrix with given Canonical Correlation Analysis components and given variances/covariances for CCA components?

So given a covariance matrix, the Canonical Correlation Analysis (CCA) components can be computed along with the correlation between corresponding pairs of CCA components. What about the other way ...
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0answers
17 views

Space is direct sum of subspaces - propostion conditions giving me problems

In Sheldon Axler's "Linear Algebra Done Right" - 2$^{\textrm{nd}}$ Edition, on the section for Direct Sums, the following proposition is stated. Following this is the proof of this 'if and only ...
0
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1answer
20 views

Why the largest singular value of a megic matrix is its magic constant?

A magic matrix is a square matrix such that the sums of the elements of each row, each column and diagonal equal to a same number, the magic constant. As reported ...
4
votes
1answer
32 views

computation involving exterior $2$-form on $\mathbb{R}^n$

Let $$\theta = \sum_{i=1}^{n-1} x_i \wedge x_{i+1}$$be an exterior $2$-form on $\mathbb{R}^n$, and $A, B \in \mathbb{R}^n$ are vectors$$A = (1, 1, 1, \dots, 1),\text{ }B = (-1, 1, -1, \dots, ...
2
votes
7answers
89 views

Prove that a matrix equals to its transpose

Let $A$ be a $(n\times n)$ matrix that satisfies: $AA^t=A^tA$ Let $B$ be a matrix such that: $B=2AA^t(A^t-A)$ Prove/disprove that: $B^t=B$ I started with: $$\begin{align} B &=2AA^t(A^t-A) \\ ...
0
votes
1answer
23 views

If the characteristic polynomials of $A$ and $B$ are equal, why are the corresponding coefficients of $\lambda^{n-1}$ equal?

Theorem: Similar matrices have the same trace. Proof: Let $A$ and $B$ be similar matrices. Then there is $P$, such that $B = P^{-1}AP$. Given that we have similar matrices then we also have ...
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0answers
28 views

Linear ODE and Fourier Series

Let $m,k_0,k$ be positive real numbers and $x_1$, $x_2$ be real-valued functions of time. Suppose we have following system of two coupled ODEs ( motivated by a coupled oscillator with two masses ...
3
votes
2answers
190 views

Is it possible that there isn't a linear span which precisely spans a vector space?

In the assignment I'm asked to decide whether given: $S = \Bigg \{ \begin{bmatrix}a &b \\ c &d\end{bmatrix} \in M_2(\mathbb{R}) \; | \; ad = 0 \Bigg \},\mathbb{F} = \mathbb{R}$. $S$ is a ...
6
votes
3answers
117 views

Lang's Linear Algebra: what's next?

I've completed the study of Lang's Linear Algebra ($3^\text{rd}$ edition). To put it simply, I have enjoyed the subject and I would like to know "what's next". In other words, I would like to know ...
3
votes
1answer
26 views

Find the area of a subset of $\mathbb{R}^3$ given by an implicit relation.

Let x, y, z be real numbers and let $A = \begin{bmatrix} 1&x&x^{2} \\ 1&y&y^{2} \\ 1&z&z^{2} \end{bmatrix} $ Let S be the subset of $\mathbf{R}^{3}$ given by $S = \{ ...
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votes
0answers
16 views

Is Frobenius norm of a gram matrix convex [on hold]

Suppose $X \in \mathbb{R}^{m \times n}$ and $S \in \mathbb{R}^{m \times m}$ Is the function $f(X) = ||XX^T - S||^2_{fro}$ convex w.r.t X ? Here, $S$ is a constant matrix. One can think of $XX^T$ as ...
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3answers
55 views

Intuitive transition from matrices to tensor-concept

I would like to know how to build intuition for the concept of a tensor using the following reasoning: If I conceive of a vector as an extension of the scalar concept, i.e. an $N \times 1$ "array of ...
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2answers
25 views

What row-operations allow this $\operatorname{Mat}_{2\times2} (\mathbb{R})$

$$ A = \begin{pmatrix} 1 & r \\ s & 1 \\ \end{pmatrix} \Rightarrow \begin{pmatrix} 1 & r \\ 0 & 1-s \cdot r \\ \end{pmatrix} = B \quad\quad r,s \in \mathbb{R} $$ Matrix B is ...
0
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1answer
19 views

Hermitian Matrix Inequality

If we have {$A_{ij}\}_{n*n}$ a Hermitian matrix. v=($v_1,v_2..v_n$), w=($w_1,w_2...w_n$) are two complex vectors. Then how can I show the inequality |$\sum_{i,j=1}^nA_{ij}v_i\overline{w_j}$|$\leq ...
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4answers
77 views

Derivative of a quadratic form

There is a Hermitian matrix $X$ and a complex vector $a$. I know that $a^HXa$ is a real scalar but derivative of $a^HXa$ with respect to $a$ is complex, $$\frac{\partial a^HXa}{\partial a}=Xa^*$$ Why ...
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2answers
28 views

simple moving average related to a mean

Am I right in this statement? Given a series of numeric values that represent measurements (y) over time (x), the closer a simple moving average is to the mean the less volatility in (y) ?
0
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0answers
24 views

Subspaces of vector space and their spans

$\def\sp{\operatorname{sp}}$ Given 2 subspaces of V, and $T\subseteq \sp(K)$ and $K\subseteq \sp(T)$ then $\sp(K)=\sp(T)$? If $ T\subseteq \sp(T)\subseteq \sp(K)$ and $K\subseteq ...
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2answers
35 views

Proof for pythagoras theorem

Let $f,g$ orthogonals to each other. $${\left\| {f + g} \right\|^2} = \left<f,f\right>+\left<g,f\right>+\left<f,g\right>+\left<g,g\right> = {\left\| f \right\|^2} + {\left\| g ...
2
votes
1answer
23 views

Find a hyperplane not intersecting $S$

I am struggling with the following problem: Let $K$ be an infinite field, $V$ an $n$-dimensional $K$-vector space, $S \subset V$ a finite subset with $0 \notin S$. Prove that there exists a subspace ...
2
votes
3answers
29 views

What ring-sum of vector spaces can possibly mean?

I'm given this test assignment, and I can't decipher what it says. Would you kindly help me? Here's the assignment itself: Let $U$ and $W$ be sub-spaces of the linear vector space $V$ s.t. $U ...
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votes
1answer
57 views

Inequality on matrix norm: $ \lVert A^n \rVert \leq \lVert A \rVert^n $ [on hold]

If $A$ is a $n \times n$ matrix and assume we have a matrix norm $\lVert \cdot \rVert$. In a proof I need the following property: $ \lVert A^n \rVert \leq \lVert A \rVert^n $. I don't know how to ...
1
vote
2answers
45 views

Jordan form of a matrix

Let $$A = \left( {\matrix{ 0 & 1 & 0 & 0 \cr 0 & 0 & 2 & 0 \cr 0 & 0 & 0 & 3 \cr 0 & 0 & 0 & 0 \cr } } \right)$$ The ...
0
votes
1answer
26 views

Proving $\mathbf x$ is a $n\times 1$ vector and $\mathbf A$ an $n\times n$ matrix, then $\mathbf x'\mathbf A \mathbf x = \text{tr} (\mathbf {Axx}')$

If $\mathbf x$ is a $n\times 1$ vector and $\mathbf A$ an $n\times n$ matrix, then $\mathbf x'\mathbf A \mathbf x = \text{tr} (\mathbf {Axx}') (\mathbf A'=transpose A) $