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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Solve the equations $\|Av\|=1/\|A^{-1}w\|$, $\|w\|=1$

I'm sorry if my question is rather stupid, but I have a brainfreeze right now. I want to prove that, for every $A\in GL(2,\mathbb{R})$ and for every $v\in \mathbb{R}^2$, $\|v\|=1$, I can find $w\in \...
16
votes
4answers
771 views

Avoiding the Cayley–Hamilton theorem [duplicate]

Every $n\times n$ matrix satisfies a polynomial equation of degree at most $n^2$, simply because the space of $n\times n$ matrices has dimension $n^2$. By the Cayley–Hamilton theorem, every matrix ...
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1answer
106 views

Linear Independence of Pre-Image

If $t:V \to W$ is a linear transformation from vector space V to W and A is a subset of V then it seems that if $t(A)$ is linearly independent in W then A is linearly independent in V. See for example ...
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2answers
22 views

Computing the projection vector $\underline{p}$ given two vectors.

Basically i'm given this: $b = \begin{smallmatrix} -2\\4\\3 \end{smallmatrix}$ and $a = \begin{smallmatrix} 1\\3\\0 \end{smallmatrix}$ I need to project vector $b$ onto the line $a$. It's also ...
5
votes
1answer
163 views

Finding the Jordan Canonical form of a $6 \times 6$ matrix

Find the Jordan Canonical Form of the following matrix $$\begin{bmatrix} 1 & 0 & 0 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 0 & 0\\ 1 & 0 & 1 & 0 & 0 &...
1
vote
1answer
24 views

Field Extensions of Q by radicals

Is Q(√6) = Q(√3,√2)? I understand that the degree of these field extensions comes from the degree of the minimal polynomial and (alternatively) basis of the field extension. I know that the basis of Q(...
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1answer
46 views

Is it true that $A = QQ^{T}$, $Q$ a square matrix, is invertible if and only if $A$ is positive definite?

I know that $A = QQ^{T}$ ($A$, $Q$ square matrices) is positive definite if and only if $Q$ is invertible for every choice of $Q$. Since the product of invertible matrices is invertible, would it be ...
4
votes
2answers
505 views

Help in finding the Jordan canonical form of a matrix

Determine the Jordan Canonical Form of the following matrix: $$A=\begin{bmatrix} 1 & 2 & 3\\ 0 & 4 & 5\\ 0 & 0 & 4\\ \end{bmatrix}$$ I am trying to determine the Jordan ...
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1answer
78 views

Suppose $T$ is a linear operator $(V,V)$, and $T^2=T$. Does this means it is the identity operator?

Since we're saying $T^2=T$, isn't $T$ a square root of itself? So that would only happen if $T=I$, correct? Another thing I tried is, for $u$, $v$ in $V$, where $Tu=v, TTu=Tu \to T(v)=v \to T=I$ ...
2
votes
1answer
28 views

Comparison of two sets of eigenvalues

Let $A$ be a symmetric matrix in $\mathbb{R}^{n\times n}$. Denote its eigenvalues by $\lambda_1\leq\ldots\leq\lambda_n$. Let $B$ be a $(n-1)\times(n-1)$ matrix induced from $A$ by taking the left-...
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3answers
1k views

Is the zero vector in the definition of linear dependence arbritary?

The definition of linear dependence according to wikipedia is The vectors in a subset $S=(v1,v2,...,vk)$ of a vector space $V$ are said to be linearly dependent, if there exist a finite number of ...
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2answers
30 views

Another Line Equation Case

At Line $L_{1}$ has equation $r = \begin{pmatrix} -5\\ -3\\ 2 \end{pmatrix} + \lambda \begin{pmatrix} -1\\ 2\\ 2 \end{pmatrix}$ A line $L_{2}$ passing through the origin intersects $L_{1}$ and ...
1
vote
1answer
481 views

Using linear algebra (e.g. matrix) methods to solve a system of linear inequalities

Say we have the equation $Ax>b$, where $A$ is an M-by-N matrix, $b$ is a known vector of length N, x is an unknown vector of length N, and the inequality sign means that each element of $Ax$ is ...
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0answers
19 views

Vector Line Equation

Consider the lines $l_{1}$ and $l_{2}$ defined by equation $ax+\left ( a-1 \right )y = 0$ and $bx+by=1$, respectively where $b\neq 0$. Find the coordinate of the intersection point of the lines $l_{1}...
2
votes
1answer
38 views

Inner Product: prove that $\langle w, v+v' \rangle = \langle w, v \rangle + \langle w, v' \rangle$

We could use linearity in the first argument, homogeneity in first argument, and conjugate symmetry properties of the dot product. So this was my attempt at proving this: We know that $\langle v+v',...
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votes
1answer
76 views

Find the equation of a budget hyperplane in R4, from an endowment point and a price vector [closed]

http://imgur.com/ofDS6MO Its been a while since I had to deal with vectors, if someone could help me along with 1.1 that would be very much appreciated
0
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1answer
39 views

How do I find such matrices $X_{1},\ldots,X_{9} \in \mathrm{M}_{2}(\mathbb{Z}) $?

Is there someone who can give at a least an idea for solving this problem? Determine the matrices $ X_{1} , X_{2} , ..., X_{9} \in \mathrm{M}_{2}(\mathbb{Z})$ such that: $$(X_{1})^{4} + \...
0
votes
1answer
72 views

When are powers of square matrices linearly independent?

If I have an n-by-n matrix $A$, is $1, A, A^2,...,A^{n^2}$ always linearly independent?
0
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1answer
20 views

Populational model with parameter

In preparation of my math exam I was trying to solve some exercises about population models. The question is as follows: Consider a population model with two species $X$ and $Y$. The populational ...
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2answers
140 views

The perpendicular distance from the origin to point in the plane

The plane $3x-2y-z=-4$ is passing through $A(1,2,3)$ and parallel to $u=2i+3j$ and $v=i+2j-k$. The perpendicular distance from the origin to the plane is $r.n = d$ but how to determine the point (...
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1answer
25 views

Associated matrix to operator on infinite dimensional spaces.

A linear operator on a vector space has a basis through which write its associated matrix. This is certainly true for finite dimensional spaces. But is it still true for infinite dimensional spaces? I ...
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1answer
60 views

Find $L\left( \left[\begin{matrix} 3 \\ -1\end{matrix} \right]\right)$?

I have the following linear map: $L: \mathbb{R}^2 \rightarrow \mathbb{R}^2$. Now, suppose $L\left( \left[\begin{matrix} 1 \\ 1\end{matrix} \right]\right) = \left[\begin{matrix} 1 \\ 4\end{matrix} \...
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0answers
61 views

Linear independence of real powers of x

I know that integer powers of x are linearly independent. I would expect that fractional powers of x (eg. $x^{n/2}$) are also linearly independent. But what about real powers of x? If I could use any ...
2
votes
1answer
67 views

Quotient group and adjoint matrix

The exercise 1211 in "Problems and Solutions in Mathematics" by Ta-Tsien: Let $M$ be an $n \times n$ matrix of integers. Suppose that $M$ is invertible when viewed as a matrix of rational numbers. ...
5
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0answers
76 views

An inner product on the dual space of a non-complete inner product space?

As is well known, for any Hilbert space $V$, there is a natural inner product on the continuous dual. (the space of all continuous linear functionals). Is there a way to endow an inner product on ...
0
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1answer
43 views

Dimension of the image and the kernel of f

I've got the linear map: f: $\mathbb{R^4} \rightarrow \mathbb{R^3}$ with $ \left( \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ x_4 \end{array} \right) \mapsto \left( \begin{array}{c} x_1+2x_2+x_3 \\ 2x_1+...
2
votes
1answer
42 views

Linear System - Laplace - Determinant

Can somebody help me? I need to find the determinant of the related matrix with Laplace's method. What is the easiest way to find it? $x+y-z+w=1\\ x+2y+z-w=-1\\ y+2z-2w=-2\\ kx+3z=0$ Thank you for ...
0
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0answers
138 views

Looking for Linear Algebra resource (problems and solutions)

I work for a small research company, and one of my bosses wants me to teach him how to use the LAPACK libraries for Fortran. I have never used Fortran or LAPACK, so I am looking for a book of linear ...
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1answer
41 views

If we know the $A A^T $'s bounded , what is the $A^T A$'s bound?

If I have a matrix $A\in \mathcal{R}^{m\times n}$, and it satisfies $\underline{a}^2 I\leq A A^T\leq \overline{a}^2 I$, can we conclude that the matrix $A^T A$ is bounded, too? and what's the bound? ...
2
votes
0answers
33 views

Linear combination of matrix elements

Consider the following sequence of problem: With $A \in \mathbb{R}^{n \times m}$, $m>n$, and $x \in \mathbb{R}^m$, I am looking to linearly combine (non-trivially) the elements of the vector $Ax$ ...
5
votes
0answers
96 views

Product of reduced row-echelon matrices is also reduced row-echelon

Show that the product of two reduced row-echelon matrices is also reduced row-echelon. That's what I think: A reduced row-echelon matrix has columns like $e_1 =(1, 0, \cdots , 0)^T$ and $e_2 =(0,...
2
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0answers
34 views

Distribution of $\langle A,x\rangle\langle A,y\rangle + \langle B,x\rangle\langle B,y\rangle$ given $\langle x, y\rangle$

Let $A$ and $B$ be independent, normal distributed $N(0,1)$ normalized unit vectors, and let $x$ and $y$ be unit vectors with given inner product $\langle x, y\rangle=u$. Can we write the ...
6
votes
1answer
198 views

Given a vector space with two inner products, there is a linear transformation taking one to another

I am looking for some hint to the following question: Let $V$ be an $n$-dimensional real inner product space and let $\langle x,y\rangle$ and $[x,y] $ both be two different inner products on V. ...
0
votes
1answer
38 views

Finding the Jordan Decomposition of a given matrix

Given $$ \begin{bmatrix} 1 & 1 & 0\\ 0 & 1 & 0\\ 0 & 0 & 2\\ \end{bmatrix}$$am I need to find a Jordan Decomposition of this matrix. For this purpose I am trying to ...
5
votes
1answer
225 views

Why more than 3 dimensions in linear algebra?

This might seem a silly question, but I was wondering why mathematicians came out with more than 3 dimensions when studying vector spaces, matrices, etc. I cannot visualise more than 3 dimensions, so ...
0
votes
1answer
79 views

Finding a vector that is perpendicular to a line.

I have two position vectors $\textbf{a} = (-1,0), \textbf{b} = (1,4)$, and have found the position vector of a general point along the line joining these to points to be $\textbf{r} = (-1+2 \alpha,4\...
0
votes
1answer
74 views

Are linear transformations $T_1$, $T_2$ invertible? [duplicate]

Let $T_1, T_2$ be two linear transformations from $\mathbb{R}^n$ to $\mathbb{R}^n$. Let $\{ x_1, x_2,....x_n\}$ be a basis of $\mathbb{R}^n$. Suppose that $T_1 x_i \neq 0$ for every $i= 1,2,...,n$ ...
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0answers
383 views

Prove if A and B are skew symmetric then A+B is skew symmetric

If $A^T$ = $-A$ which means A is skew symmetric then prove that $(A+B)$ is also skew symmetric. I managed to prove it like this: $(A+B)^T$ = $A^T$+$B^T$ =$(-A+-B)$=$-(A+B)$ Therefore $(A+B)^T$=$-(A+...
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1answer
82 views

Minimize $w=9y_1+4y_2$ subject to linear inequalities

Minimize $w=9y_1+4y_2$ subject to : $4y_1+9y_2\geq 360$ $y_1+4y_2\geq 40$ $y_1\geq 0,~y_2\geq 0$
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2answers
20 views

Stationary values

I'm not sure how to go on and answer this question, I appreciate any help. Show that $$f(x) = \ln(3x^2 - 2x -1) - 4x^2$$ has a stationary value when $x$ satisfies $$12x^3 - 8x^2 - 7x + 1 =0$$
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1answer
42 views

operator norm of a matrix and the largest eigenvalue

Is it true that for any $n \times n$ matrix $A$, with real entries, and where all eigenvalues are real, and non zero, the operator norm ( where$||A|| = \max_{|x|=1} |Ax|$ with |.| is the standard ...
2
votes
0answers
100 views

What should this definition be?

This is from Advanced Linear Algebra by S.Roman. Definition (on page 356). Referring to the figure below, let A be a set and let $\mathcal S$ be a family of sets. Let $$ \mathcal F = \{g\colon A\...
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0answers
26 views

A question on bounded operator $\|T_A x\|\leq K\|x\|$

We consider an operator $T_A:H\rightarrow H$, where $H$ is an Hilbert Space and $A$ is its associated $N\times N$ matrix. $T_A$ is said "bounded" if there exists a constant $K>0$ such that $$\|T_A ...
1
vote
1answer
41 views

Find a function $h$ such that $g(x) =\langle f, h \rangle$

Let $P_2(\mathbb R)$ be an inner product space with $\langle f, h \rangle = \int_{0}^{1}f(t)h(t)dt$. Let $g(f) = f(0) + f'(1)$. Find $h(t)$ such that $g(f) = \langle f,h \rangle$. I tried ...
2
votes
2answers
60 views

Prob. 9, Sec. 3.9 in Erwin Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS: Finite-dimensional range and the form of images

Let $H$ be a Hilbert space, and let $T \colon H \to H$ be a bounded linear operator. Then how to show the following? The range of $T$ is finite-dimensional if and only if $T$ can be represented in ...
2
votes
2answers
94 views

Polynomial in several variables over $GF(2)$

Can anyone please explain how this Lemma has been proved? Lemma: Let $f$ be a nonzero polynomial in variables $x_1,\ldots,x_n$ over $GF(2)$, and let $d$ be the maximum degree of $f$ with respect to ...
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1answer
39 views

how to understand this identity about the range in linear algebra

I see the identity in page 48 of paper http://arxiv.org/pdf/0909.4061.pdf. Specially, if $U^TU=UU^T=I$, then we will have $U^T\text{range}(M)=\text{range}(U^TM)$, where $\text{range}(M)$ means the ...
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votes
3answers
2k views

If I have a diagonal matrix, is it necessarily the product of two other diagonal matrices?

We know that the product of two diagonal matrices forms another diagonal matrix, since we just multiply the entries. So my question is, does the converse necessarily hold? In other words, if I have a ...
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votes
1answer
83 views

Let $\left| {{a_{ii}}} \right| > \sum\limits_{i \ne j} {\left| {{a_{ij}}} \right|} $.Why does $A$ is nonsingular? . [duplicate]

Let $A \in {M_n}$ and $\left| {{a_{ii}}} \right| > \sum\limits_{j \ne i} {\left| {{a_{ij}}} \right|} $.Why does $A$ is nonsingular?
3
votes
2answers
86 views

If $T: V\rightarrow W$ is isomorphic then $T$ is invertible. Prove that its inverse is a linear transformation

Ok so I know $T$ is a linear transformation so it stands that this is true: $T(\alpha x + y) = \alpha T(x) + T(y)$ However, I'm a little lost on how I should use this information to prove that the ...