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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Maple: How do I type “solve” with an arrow under?

I am trying to learn using Maple 18 (Mac). I have defined a function with a list of X and Y values. f := x->LinReg(X, Y, x) Now I would like to output the unknown "x" value that correlates with ...
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4answers
192 views

Determinant: Alternative Definitions

Reference Foundation for: Determinant: Continuity Problem Given a vector space $V$. Consider an endomorphism $T:V\to V$. The rank of an endomorphism: ...
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1answer
27 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 ...
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1answer
30 views

Preferred way to write elements of the direct sum of vector spaces

Suppose $V$ and $W$ are vector spaces over the same field and $V\oplus W$ is their direct sum. Reading through the literature I found essentially two ways of writing elements of $V\oplus W$. 1.) We ...
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18 views

Solve Karush–Kuhn–Tucker conditions

solving a constrained optimizing problem with equality constraints can be done with the lagrangian multiplier. (http://en.wikipedia.org/wiki/Lagrange_multiplier) This approach leads to a system of ...
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110 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 ...
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1answer
34 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 ...
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Show that a T-cyclic subspace is the smallest T-invariant subspace that contains an element

I am working on a problem in Linear Algbra, fourth edition, by Friedberg. Problem 11 section 5.4, page 323. I would like feed back, on my proof. In particular, in part (b), I ask a question with ...
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1answer
49 views

General form for the rotation of a function.

When rotating linear functions, I would approach the task geometrically (find invariant point etc.), yet I tried using a matrix which worked nicely. This was what I did to rotate $y=2x+1$ by ...
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1answer
494 views

Is the “Constant Rank Theorem” the same as the “Domain Straightening Theorem”? Which theorem is which?

Wikipedia says that the inverse function theorem is a special case of the "constant rank theorem". I'm pretty sure this is supposed to be the same theorem as the "Rank Theorem" on p. 47 of Boothby ...
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505 views

Find the values of $x$ which makes $\det (A)=0$ without expending determinant

Find the values of $x$ which makes $\det(A)=0$ without expending determinant: Let $A$ : $$\begin{bmatrix}1 & -1 & x \\2 & 1 & x^2\\ 4 & -1 & x^3 \end{bmatrix} $$ How can I ...
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1answer
38 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 ...
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2answers
35 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 ...
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116 views
+50

covariant and contravariant components and change of basis

I encountered the following in reading about covariant and contravariant: In those discussions, you may see words to the effect that covariant components transform in the same way as basis ...
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4answers
36 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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18 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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2answers
64 views

Combining two convolution kernels

Is it possible to combine two convolution kernels (convolution in terms of image processing, so it's actually a correlation) into one, so that covnolving the image with the new kernel gives the same ...
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0answers
25 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 ...
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2answers
34 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) = ...
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2answers
30 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 ...
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1answer
66 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. ...
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3answers
498 views

Dimension of vector space of 2x2 skew symmetric matrices

I had a question about the dimension of this subspace. This was related to a problem that had a case of n x n matrices, but I accidentally read it as the special case of 2x2, but never the less the ...
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1answer
34 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 ...
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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 ...
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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 ...
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1answer
22 views

Odd coefficient in $M\in \mathcal{M}_n(\Bbb{Z})$ satisfies $n\le m\le n²-n+1$.

Let $M\in \mathcal{GL}_n(\Bbb{Z})$ I would like to prove that all odd coefficient of $M$ satisfies $n\le m\le n²-n+1$. In fact I don't see why $m$ is necessary bigger than $n$. I can only prove ...
2
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1answer
86 views

“Internal” and “external” in maths, and also in vector spaces

I have looked at 3 books and it is clear that "internal" and "external" are two styles of defining something, I would like to know what they mean "generally" - that is very soft but it is clear to me ...
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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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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 ...
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1answer
34 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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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 ...
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0answers
29 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 ...
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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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4answers
79 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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37 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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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 ...
6
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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 ...
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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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3answers
56 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 ...
2
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1answer
1k views

what is dimension of orthogonal complement of a subspace of a vector space.

This is last part of my other question. I don't understand the last part of problem. Feel free to edit the question. c) Let $V$ be a vector space of real $n \times n $ symmetric matrices, what is ...
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0answers
19 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
59 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 ...
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29 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) ?
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2answers
89 views

What is the geometric interpretation of a vector squared?

I'm working through Introduction to Space Dynamics by William Tyrrell Thomson. I am having to do a lot of research to make it through even small parts, but I am unable to find information to make me ...
0
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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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1answer
134 views

condition for upper triangular matrix

Consider the following condition from this other post Define $S_k = \operatorname{span} (e_1, \ldots, e_k)$, where $e_i$ the standard basis vectors. Clearly, the linear map $T$ is upper triangular ...
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1answer
29 views

Count and description of vertices of certain faces of the Tridiagonal Birkhoff polytope $\Omega^t_{d+k}$

For $k \ge 1$, $d \ge 2$ and $k \le d - 1$, let ${}^f_d\Omega^t_{d+k} (d;c_k(d - 1))$ be the intersection of $k - 1$ facets of the Tridiagonal Birkhoff polytope $\Omega^t_{d+k}$ with equations: ...
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19answers
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What are some applications of elementary linear algebra outside of math?

I'm TAing linear algebra next quarter, and it strikes me that I only know one example of an application I can present to my students. I'm looking for applications of elementary linear algebra outside ...
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1answer
111 views

Exponential of a matrix and related derivative

I have $ X \in M(n,\mathbb R) $ to be fixed. I define $ g(t) = \det(e^{tX}) $ Then the author proceeds as follows: $ g'(s) = \frac {d}{dt} g(s+t) $ = $ \frac {d}{dt} \det(e^{(s+t)X}) |_{t=0} $ = $ ...
3
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1answer
31 views

Phase Portrait of DE's

How would I graph the phase portrait of $$ x' = x^2+y^2-2 \qquad y' = y-x^2 $$ ? Could someone provide some insight by hand or perhaps a computer-generated image?