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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103 views

symmetric matrix

Consider the basis $β=\{(1,1,0),(1,0,-1),(2,1,0)\}$ for $\mathbb R^3$ Does the following matrix $A=[T]_β^β$ define symmetric mappings of $\mathbb R^3$? \begin{bmatrix}-1 & 1 & 2 \\ 1 & 4 ...
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3answers
80 views

Simplify a sum of fractions

I am stuck trying to get from: $$\frac{pZ(a)}{pZ(a) - (1-p)Z(b)} - \frac{p(pZ(a) - (1-p)Z(b))}{pZ(a) - (1-p)Z(b)} $$ to $$\frac{p(1-p)(Z(a) - Z(b))}{pZ(a) - (1-p)Z(b)} $$ Obviously my problem is ...
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0answers
29 views

Inverse of $(U^H X U + D)$ where U is unitary, X and D diagonal

Given complex unitary matrix U, and full rank diagonal matrices X and D with positive entries. I'm looking for an efficient way to compute: $(U^HXU+D)^{-1}$ The matrix inversion identity doesn't ...
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1answer
175 views

Prove that if matrix $A$ is an $m\times n$ and $B$ is $n\times p$, then $\operatorname{rank} AB$ is less than or equal to $\operatorname{rank} B$

Prove that if $A$ is an $m\times n$ matrix and $B$ is an $n\times p$ matrix, then $\operatorname{rank} AB$ is less than or equal to $\operatorname{rank} B$. The hint is: prove that if the $k$th ...
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1answer
327 views

Prove that direct sum of linear transformations is a block matrix

In linear algebra, I'm facing a question I cannot quite formalize into a full proof. I want to prove that a direct sum of linear transformations is a block matrix. Here's the formal question: Assume ...
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2answers
247 views

Solve the differential equation

Consider $\frac{dx}{dt} = Ax$ where $A$ is the matrix $$ \begin{bmatrix} 1 & 0 & 1 \\ 0 & 0 & -2 \\ 0 & 1 & 0 \\ \end{bmatrix} $$ ...
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1answer
116 views

Show the stable age structure

Considering the population process described by where $γ$ is the dominant eigenvalue of $L$ $l$ denotes the survival function of the Leslie matrices and $L$ is the Leslie matrix below We are trying ...
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2answers
80 views

If $x\perp \mathrm{span}\{r_1,\dots,r_p\}$, can we prove $x\notin\mathrm{span}\{v_1,\dots,v_p\}$?

Notations: For a scalar $a\in\mathbb{R}$, denote $$\mathrm{sgn}(a)=\left\{ \begin{array}{l l} 1 & \mbox{if } a>0\\ 0 & \mbox{if } a=0\\ -1 ...
3
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2answers
135 views

Find a general control and then show that this could have been achieved at x2

Determine the general form of $u_0, u_1 ~\text{and} ~ u_2$ if a system of difference equations of the form $$x_{n+1} = Ax_n + Bu_n,$$ where: $$A = \begin{pmatrix} 3 & 2 & 2 \\ -1 ...
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1answer
375 views

Spectrum of the cycle graph $C_n$

I am trying to find out the spectrum (the collection of eigenvalues) with their multiplicities of the cycle graph $C_n$. Assuming that $X=\pmatrix{x_1\\x_2\\\vdots\\x_n}$ is the eigenvector, by ...
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1answer
252 views

LU decomposition of matrices

Although I know how the LU decomposition is done, given the following two matrices: $\begin{pmatrix} 0 & 2 & 3\\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$ and $ ...
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1answer
117 views

Population Transition Matrix

I am given a matrix $$\begin{bmatrix} 0.7 & 0.2\\ 3 & 0 \end{bmatrix}$$ which is said to be the transition matrix for a fish population. There is a 70% adult survival rate and a 20% ...
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1answer
94 views

Relation between Hadamard product and scalar product

Is there a known relation/formula for $$(A\circ B, C)$$ where $\circ$ is the Hadamard product and $(\cdot, \cdot)$ is the scalar (euclidean) product? In particular, I have a vector $y$ and a two ...
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291 views

Collinearity, coplanarity and determinant

I don't get what the question is asking me. I'm confused why they add '1's to the matrix. Anyway here's my attempt. For part (i), my analysis is that since P2, P3, P4 are not collinear, therefore ...
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1answer
122 views

Find ${T^{*}}(p(x))$ for an arbitrary $p(x) = a+bx+cx^2$

Consider the vector space $P^2(\mathbb{R})$ of real quadratic polynomials with inner product $$\left\langle p(x),q(x) \right\rangle = p(-1)q(-1)+p(0)q(0)+p(1)q(1).$$ Let $T:P^2(\mathbb{R}) \to ...
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2answers
392 views

determinant linear factorisation

there's this determinant problem I've been working on for several days now whose answer I can't quite get to: $$ D = \left| \begin{array}{ccc} a^3+a^2 & a & 1\\ b^3+b^2 & b & 1\\ ...
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0answers
36 views

Let E a projection,calculate $f (E)$

Let $V$ a vector space over the field $F$ and E be a projection of $V$ and f is an element of $F[t]$, of grade greater than or equal to 1, calculate f (E) and determine the relation between ...
4
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2answers
146 views

What are the generators for $SL_n(\mathbb{R})$ (Michael Artin's Algebra book)

The book asks you to prove that $SL_n(\mathbb{R})$ is generated by elementary (row operation) matrices in which one nonzero off-diagonal entry is added to the identity matrix. For example, $$ ...
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1answer
44 views

Show that the eigenvectors of $T$ generate a subspace of $C ^ {2}$ of dimension 1.

Let $T$ an operator on $C ^ {2}$given by $T (x_ {1}, x_ {2}) = (2x_ {1}-x_ {2}, x_ {1})$. Show that the eigenvectors of $T$ generate a subspace of $C ^ {2}$ of dimension 1. I did this: Applying ...
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1answer
81 views

Product rule for inner products using the 3 conditions

I understand there are multiple ways of of proving the product rule for the derivative of an inner product, though I cannot figure out how to do this one specifically: let $\alpha,\beta :R ...
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2answers
212 views

How could I go about proving that $\dim U = \dim V + \dim V^{\bot}$

How could I go about proving that $$\dim \ U = \dim \ V + \dim \ V^{\bot}$$ Where $U$ is a finite dim vector space. If I know that $V$ is a subspace of $U$, and let $V^{\bot}$ be the set of all ...
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1answer
108 views

norm of $T$ induced by the 1-norm

How to prove this: If $T\in L(\mathbb{R}^n;\mathbb{R}^n)$, then $\|T\|= \sup_{i \in \{1,\dotsc,n\}} \|Te_i\|_1$, for $\|T\|$ the operator norm of $T$ induced by the $1$-norm on $\mathbb{R}^n$.
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1answer
364 views

Does an Inner Product Always Induce a Metric?

I am working with linear algebra over finite fields, specifically $F_2$. In class my professor has explained that every inner product induces a norm, $\sqrt{\left < v,v \right>}$ which in turn ...
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3answers
52 views

Question about basis

I have to prove that $1, (1-t), (1-t)^2$, and $(1-t)^3$ is a basis of for $P^{3}$ but I am not sure how to start this problem. Also I have to find the coordinates of $p(t) = 1 +t^3$ Thank you for ...
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4answers
3k views

Linear dependency of polynomials question

I have to determine whether the polynomials $p_1(x)=2-x^2$, $p_2(x)=3x$, $p_3(x)= x^2 +x-2$ are linearly dependent or independent but I am not sure how to start. Anyone care to enlighten me? Also I ...
0
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1answer
60 views

Find positive and negative indecies of inertia of a quadratic form

Find positive and negative indecies of inertia of a quadratic form $ q(x)= trX^2 $ on the $M_n(\mathbb{R})$ (vector space of square matrices ). How to do that? Thanks.
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2answers
96 views

Finding a linear equation

I cannot understand this question: "Find a linear equation (and parametrics) to $v$ where $v$ is perpendicular to the line segment of the extremes $(1,2,1)$ and $B$ $(1,8, -5)$, dividing it in half." ...
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41 views

Does this imply $A+B\sim D_A+D_B?$

Let a symmetric matrix $A$ be similar to diagonal matrix $D_A$ and a symmetric matrix $B$ be similar to diagonal matrix $D_B.$ Does this imply $A+B\sim D_A+D_B?$
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2answers
140 views

Linear ODE repeated eigenvalues how to find more than 2 generalized eigenvectors

So I've searched around the web for a few hours now, as (i) $\mathbf A = \begin{pmatrix}2&1\\0&2\end{pmatrix}$ The characteristic polynomial is $(\lambda-2)^2=0$, so $\lambda=2$, repeated. A ...
4
votes
4answers
209 views

The matrix I mentioned below is irreducible and primitive or isn't?

$$ \left[ \begin{array}{@{}ccccc@{}} 0.9& 0.1& 0& 0& 0& 0& \\ 0& 0.9& 0.1& 0& 0& 0& \\ 0& 0& 0.9& 0& 0& 0.1& \\ 0& 0& ...
4
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1answer
67 views

How to compute determinant of $A$ such that $A=(I+\ [c_ic_j])\in M_n(\mathbb R)$ ,$c_i\in\mathbb R$

assume $\ [c_ic_j]_{n\times n}\in M_n(\mathbb R)$ such that $c_1,c_2,\ldots,c_n\in\mathbb R$ and $I$ be identity matrix how compute $$\det (I+\ [c_ic_j])=?$$ Thanks in advance
23
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7answers
2k views

Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix

I am looking for an intuitive explanation as to why/how row rank of a matrix = column rank. I've read the proof at http://en.wikipedia.org/wiki/Rank_of_a_linear_transformation and I understand the ...
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2answers
119 views

Characteristic Polynomial Question

There is a question that I encountered: to find a polyomial $g(t)$ such that linear map $g(T)$ has characteristic polynomial $p(t) = (11-t)^r(17-t)^{n-r}$ Obviously the eigenvalues are $11$ and $17$ ...
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2answers
89 views

Prove that $S\circ T$ and $T\circ S$ have the same characteristic polynomial.

Please help me with this question: Let $S$ and $T$ be linear operator on the same finite dimensional vector space. Suppose $S$ is invertible. Prove that $S\circ T$ and $T\circ S$ have the same ...
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1answer
293 views

Where are the resources for a collection of linear algebra problems?

Are there any resources consisting of a collection of problems on linear algebra for students to practice? I am looking for good interesting problems which test students’ understanding. These ...
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1answer
158 views

Alternative Almost Complex Structures

Let $V$ be a real vector space. An almost complex structure on $V$ is a map $J : V \to V$ such that $J^2 = -\mathrm{id}_V$. An almost complex structure gives $V$ the structure of a complex vector ...
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3answers
44 views

Basis and dimension

Let $V=\mathbb{R}^2$ and $W_1=\{(a_1,0):a_1\in \mathbb{R}\}$. Given examples of two different subspaces $W_2$ and ${W_2}'$ such that $V=W_1 \bigoplus W_2$ and $V=W_1 \bigoplus {W_2}'$.
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1answer
726 views

Example of a Markov chain transition matrix that is not diagonalizable?

It is well-known that every detailed-balance Markov chain has a diagonalizable transition matrix. I am looking for an example of a Markov chain whose transition matrix is not diagonalizable. That is: ...
0
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1answer
46 views

Let $S$ be a linear operator on $W=P_3(\mathbb R)$ defined by $S(p(x)) = p(x) - \frac {dp(x)} {dx}$

Let S be a linear operator on W = $P_3(\Bbb R)$ defined by S(p(x)) = $p(x) - $$\frac {dp(x)} {dx}$. (a) Find nullity (S) and rank (S) (b) Is S an isophormism? If so, write down a formula for ...
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76 views

Why does the map $x^2$ have constant rank?

I'm just trying to wrap my head around the rank of a map via some examples. Now, if I have the smooth map of manifolds $F:\mathbb{R} \to \mathbb{R}, F(p) = p^2$, then the differential is given by ...
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1answer
76 views

Proofs about invertible linear functions

Let $G\subset L(\mathbb{R}^n;\mathbb{R}^n)$ be the subset of invertible linear transformations. a) For $H\in L(\mathbb{R}^n;\mathbb{R}^n)$, prove that if $||H||<1$, then the partial sum ...
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2answers
673 views

How to make a matrix positive semidefinite

We have a symmetric matrix $A$, with some entries specified and others not. We are trying to find the values of the unspecified entries so that the matrix $A$ becomes positive semidefinite. How can I ...
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1answer
93 views

Sums and Direct Sums

My text says: $$ U + W = \{(x,y,0):x,y ∈ F\}$$ if $$U = \{(x,0,0) \in F^{3} : x ∈ F\}$$ $$W = \{(0,y,0) \in F^{3} : y ∈ F\}$$ I have to verify this. What I did was add $(x,0,0)$ and$ (y,0,0)$ ...
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1answer
1k views

Linear transformation applied to a multivariate Gaussian random variable - what is the mean vector and covariance matrix of the new variable?

Given a random vector $\mathbf x \sim N(\mathbf{\bar x}, \mathbf{C_x})$ with normal distribution. $\mathbf{\bar x}$ is the mean value vector and $\mathbf{C_x}$ is the covariance matrix of ...
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2answers
103 views

Let $f$ be a polynomial with real coefficients and $A$ a symmetric matrix of $n\times n$ with elements in $\mathbb{R}$. Prove that $f(A)$ is symmetric

Let $f$ be a polynomial with real coefficients and $A$ a symmetric matrix of $n\times n$ with elements in $\mathbb{R}$. Prove that $f(A)$ is symmetric. Suppose $A$ is hermitian and that $f$ has ...
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2answers
438 views

Determining values for an vector entry to make vectors independent.

Determine all values of $h$ such that the vector set $\{ (3, 1, 2), (0, 1, -1), (3, 3, h) \}$ is independent.
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1answer
36 views

Show that the set $B = {T: F ^ {n} \rightarrow F ^{m} | p = 1, .., m, q = 1, …, n} $. is basis

Let $F$ a field and $V$ the vector space of all transformations $ T^{pq}: F ^{n}\rightarrow F ^{m} $. Show that the set $B$ is a basis of $V$; here $B = {T: F ^ {n} \rightarrow F ^{m} | p = 1, .., ...
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3answers
95 views

Showing a Linear Transformation is $0$ through an Inner Product Condition

$\fbox{Setting}$ Let $V$ be an inner product space and $\tau$ a linear transformation on $V$. Suppose that $\exists w \in V$ s.t. $\forall u \in V$, $\left\langle \tau(u),w \right \rangle = 0$. ...
0
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2answers
857 views

Vector (Linear Combination)

How do you solve this problem? Write each vector as a linear combination of the vectors in S if possible: $S = \{(2,0,7),(2,4,5),(2,-12,13)\}$ $u = (-1,5,-6)$ I only got to the point where left ...
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1answer
66 views

A bound on the norm of the sum of two index-disjoint matrices

Given two matrices, it is well known that $\parallel A+B \parallel _2 \leq \parallel A \parallel _2+\parallel B \parallel_2$. Now, suppose that the nonzero indices are disjoint (i.e., $A$ is nonzero ...