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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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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129 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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2k 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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52 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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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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53 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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82 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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56 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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75 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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178 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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53 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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213 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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426 views

If the Wronskian is zero at some point, does this imply 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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97 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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52 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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38 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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1answer
84 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
72 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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40 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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54 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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144 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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51 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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58 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
69 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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222 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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106 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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76 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
152 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
54 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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199 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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2answers
61 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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3answers
76 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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141 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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53 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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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
25 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
22 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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45 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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41 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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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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119 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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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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14 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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66 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 $1 \le d_l \le 3$ ...
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1answer
63 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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1answer
80 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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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 ...
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196 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 ...
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1answer
97 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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240 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 ...