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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Can the gradient be expressed with contravariant components?

I read that the gradient is an example of a quantity that transforms covariantly since in the below expression for the gradient $$\frac{\partial x^j}{\partial x'^i}$$ appears instead of ...
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24 views

Structure of skew $0,\pm1$ matrix

Let $S$ be a skew symmetric matrix with entries from $\{-1,0,+1\}$. We know that there is a unitary matrix $U$ that diagonalizes $S$. There are only $3^{\frac{n^2-n}{2}}$ such matrices and still lower ...
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Components of vector in dual basis transform covariantly

I am trying to understand how components of a vector in the dual basis transform covariantly as mentioned in this quote. If you seek to define a quantity (such as vector A) that remains ...
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46 views

A question on matrices

Let $M\in\Bbb \{0,1\}^{n\times n}$ be a rank $t\leq n$ matrix and we know that it can be rewritten as $A+B$ where $A$ is $\{0,1\}$ symmetric and $B$ is $\{-1,0,+1\}$ skew symmetric. Are there any ...
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1answer
30 views

Determining a basis for a space of polynomials

Determine a basis from the following set of second degree polynomials. Does this basis span the space of the second degree polynomials? What is the dimension of the (sub)space that it spans? ...
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38 views

To show that $\operatorname{Rank}(\mathbf{A}-\mathbf{I})=\operatorname{Nullity}(\mathbf{A})$

Problem is: Let $\mathbf{A}$ be $n\times n$ matrix with real entries such that $\mathbf{A}^{2} = \mathbf{A}$. If $\mathbf{I}$ denotes the identity matrix, then how do I prove the result: ...
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1answer
65 views

Powers of adjacency matrix doesn't seem to correspond to observed number of paths on graph

I would really appreciate some help on this! $A^n$ represents $n^{th}$ power of the adjacency matrix of a graph. I keep reading that the $A^n_{ij}$ entry equals "the number of paths of length n ...
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3answers
25 views

If $T : F^{2 \times 2} \to F^{2\times 2}$ is $T(A) = PA$ for some fixed $2 \times 2$ matrix $P$, why is $\operatorname{tr} T = 2\operatorname{tr} P$?

I am asked to prove that if $T$ is a linear operator on the space of $2 \times 2$ matrices over a field $F$ such that $T(A) = PA$ for some fixed $2 \times 2$ matrix $P$, then ...
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2answers
29 views

Whether a set of vectors span a subspace that includes a given vector

Do the vectors $(0, 1, 2), (1, 2, 1), ( -1, 2, 4)$ a) span $\mathbb R^{3}$ b) span a subspace that includes $w = (-2, 2, 10)$ I know they don't span $\mathbb R^3$ since they are ...
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1answer
35 views

Proof If $A_{2x2}$ and $\lambda$ real number then $|\lambda I-A|=\lambda^2-(\operatorname{tr}A) \lambda+|A|$

I have this problem : Proof If $A_{2x2}$ and $\lambda$ real number then $|\lambda I-A|=\lambda^2-(\operatorname{tr}A) \lambda+|A|$ This is what I did : I took an arbitrary $A$ $$ A= \left( ...
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66 views

How do I solve the following equations?

I have the following problem: \begin{align} Y &= A X \\ Y &= R \exp \left(j \Phi\right) \text{element-wise}\\ X, R, \Phi &\in \Bbb R \\ A, Y &\in \Bbb C \end{align} I know what A is, ...
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4answers
33 views

You get to choose any two non-zero vectors $v ∈ \mathbb{R}^3$ and $u∈\mathbb{R}^3$. How can I find an equation for a plane?

How do I find the equation of the plane or the line that is spanned by the vectors $u$ and $v$. I just don't understand what the question means when they ask me to find the equation.
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2answers
49 views

Is a linear operator applied to itself still linear?

I'm not sure if this is a really obvious result, but I'd like to make sure I've not got it wrong. Here is my attempt: Take the vector space V to be a (complex) Hilbert Space. For $x,y$ in $V$ and ...
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1answer
49 views

Finding non-zero eigenvalues of a $5\times 5$ matrix [on hold]

Product of the non-zero eigen values of the matrix ? ...
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38 views

Perturbation theory of the eigenvalues about the symmetric matirx

From Weyl's theorem, i.e.: Let $A$ and $E$ be $n\times n$ real symmetric matrices. Let $\alpha_1\geq\ldots\geq\alpha_n$ be the eigenvalues of $A$ and $\hat{\alpha}_1\geq\ldots\geq\hat{\alpha}_n$ be ...
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26 views

LU factorization of a modify matrix

Suppose you know $L$, $U$, decomposition LU of a matrix $M+I$ ($M+I=LU$). Lets $J$ a diagonal matrix whose elements are $0$ or $1$. Is there any relation between the factorization LU of $M+I$, and ...
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3answers
124 views

Examples of combinatorial/probabilistic proofs of theorems in linear algebra

Are there any examples of combinatorial/probabilistic proofs of theorems in linear algebra? Motivation: I see here, the inverse is true.
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21 views

a doubt about the wronskian and linear dependency of functions

We know that if for functions $f$ and $g$, the Wronskian $W(f,g)(x_0)$ is nonzero for some $x_0$ in [a,b] then f and g are linearly independent on [a,b]. If f and g are linearly dependent then ...
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1answer
27 views

Prove dimension of sum of two subspaces

Let $U$ and $W$ be subspaces of $\mathbb{R^n}$ where $\dim(U)=n-1$, $\dim(W)=n-3$ and $n\geq 3$ Prove that $\dim(U\cap W)\geq n-3$ I used the property that both $U$ and $W$ are subspaces of ...
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43 views

Eigenvalues of $\frac{1}{2}(A+ A^T)$ [on hold]

If we know the eigenvalues of $\frac{1}{2} (A+A^T)$ with $A$ a real $m \times m$ matrix, what can we say about the eigenvalues of $A$?
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28 views

Positive dot products and special linear dependence

I would be very happy if there would be a short proof of the following fact: Let $w, w_1, \dots, w_k$ be vectors in ${\mathbb{R}}^n$ with rational entries. Suppose that for every $v \in ...
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1answer
20 views

Show that this mapping (with respect to basis) is a linear transformation.

Let T be a linear transformation from Rn to Rn. Let B = {b1, b2,...bn} be a basis of Rn. Show that the map taking [v]B to [T(v)]B is a linear transformation from Rn to Rn. This linear transformation ...
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21 views

Differentiating a matrix function with respect to a scalar

I would like to differentiate the following with respect to psi (partial): $$ \operatorname{trace}\bigl((X^\top X)^{-\psi} P\bigr). $$ Here we have that: $ X \in \mathbb{R}^{p \times n}, P \in ...
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2answers
49 views

Determine which set span $\mathbb{R^3}$

Let $v_1,v_2,v_3$ be vectors in $\mathbb{R^3}$ such that $\langle v_1,v_2,v_3\rangle=\mathbb{R^3}$ Determine which of the following sets span $\mathbb{R^3}$ i)$S=\{v_1,v_2\}$ ...
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16 views

SVD and base changes matrices

I'm not hugely comfortable with linear algebra, so wanted to double check that the following reasoning was correct. Does it hold that, given two matrices R and B $U R B U^T=U R U^T U B U^T= U R U^T ...
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24 views

Prove the Basis of Column Vectors

Let $V$ be a vector space over field $\mathbb{F}$. Let $B=\{u_1,...,u_n\}$ be an ordered basis of $V$. Show that $\{[u_1]_c,...[u_n]_c\}$ is a basis of $M_{n,1}(\mathbb{F})$ for every ordered basis ...
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22 views

Find optimal matrix to maximize the expected expression

I am interested of the following function with $Q$. $$f(Q)=h^TQh-\frac{1}{(h^TQh+1)^2}-g^TQg+\frac{1}{(1+g^TQg)^2}$$ where $h$ and $g$ are both given $N\times 1$ vectors. And $Q$ is $N \times N$ ...
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1answer
36 views

Prove an upper bound for the determinant of a matrix A

Let $A$ be a $3 \times 3$ real matrix with all $0\le a_{ij} \le 1$. Show that $\det(A) \leq 2$ and find such matrices with $\det(A) = 2$. Let $A$ be a $n \times n$ matrix with all $0\le a_{ij} \le ...
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2answers
41 views

Prove that U and W are contained in U+W

Let $V$ be a vector space and $U\leq V,W\leq V$ Prove that $U$ and $W$ are contained in $U+W$. I do not understand the requirement of the question. Is it prove that $U,W$ is the subset of $U+W$ or ...
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1answer
34 views

Space of continuous functions linear operator eigenvalues

Let $V$ be the vector space of continuous functions from $\mathbb R$ to $\mathbb R$. Let $T$ be the linear operator on $V$ defined as $$(Tf)(x)=\int_0^x f(t)\,dt$$ Prove that $T$ doesn't have ...
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1answer
40 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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1answer
33 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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2answers
46 views

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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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
44 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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1answer
39 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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1answer
32 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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5answers
173 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
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2answers
38 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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4answers
43 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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19 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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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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37 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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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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1answer
17 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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21 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
35 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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2answers
32 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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25 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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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: ...