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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2
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1answer
26 views

Uncountable Basis?

I was reading up on the difference between countable and uncountable sets, and was wondering if there was a basis of uncountable size. I now know there are, however they all seem to be covering ...
0
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1answer
30 views

Interpretation of $(r,s)$ tensor

A tensor of type $(r,s)$ on a vector space $V$ is a $C$-valued function $T$ on $V×V×...×V×W×W×...×W$ (there are $r$ $V$'s and $s$ $W$'s in which $W$ is the dual space of $V$) which is linear in each ...
1
vote
1answer
40 views

How to solve the equation $AX=B$ in Matlab?

I am trying to solve an equation of the form AX=B where A, X and B are following matrices. I have the A and B matrices and I have to find the value of matrix X. How can I find the value of matrix X. I ...
2
votes
2answers
31 views

If the 2-norm of a matrix is small, the trace of the matrix is also small

Is it true that If the 2-norm of a symmetric real matrix is small, then the trace of the matrix is also small? I played around with some matrices in MATLAB and discovered this phenomenon. Does there ...
0
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1answer
44 views

Volume of Region in $\mathbb{R}^2$.

Consider $$ S = \left\{(x,y) \in \mathbb{R}^2; -N-\frac{1}2 \le x \le N + \frac{1}2, |\alpha x-y| \le \frac{1}N \right\}$$ where $N \in \mathbb{N}, \alpha \in \mathbb{R}$. I'm having a hard time ...
-1
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1answer
52 views

Given linear maps $T:V\to W$ and $S:V\to W$ does there exist linear map $F:V\to W$ with ker F=ker $T\cap $ ker S

Given linear maps $T:V\to W$ and $S:V\to W$ does there exist a linear map $F:V\to W$ with ker F=ker $T\cap $ ker S, where $V$ and $W$ are different vector spaces? What if $V=W$? The answer is in ...
2
votes
1answer
82 views

Are all fields vector spaces?

Are $\mathbb{Z_p},\mathbb{Q},\mathbb{R},\mathbb{C}$ above themselves vector space? Is a field above anoother field a vector space? As for 1. we know that $\Bbb R^n$ is a vector space so in ...
0
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3answers
30 views

Find a value r so that the vector v is in the span of a set of vectors

Find the value r so that, $$v = \begin{pmatrix} 3 \\r \\-10\\14 \end{pmatrix}$$ is in the set, $$ S= \text{span}\left(\begin{pmatrix} 3\\3\\1\\5 \end{pmatrix}, \begin{pmatrix} 0\\3\\4\\-3 ...
1
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2answers
42 views

Find $2\det ( \frac{1}{2} A )$ given that $A$ is $3\times 3$ and $\det(A)= -2$

Here is a question that should be done today: If $A$ is $3\times 3$ and $\det(A)= -2$, find $2\det(\frac{1}{2}A)$. I solved this problem but I am not sure because the way I used is not accurate! ...
1
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1answer
40 views

Can I find the minimal polynomial by using the characteristic polynomial?

Let's say I have the characteristic polynomial of an operator: $$p(z)=(z-\lambda_1)^{j_1}(z-\lambda_2)^{j_2}\dots(z-\lambda_n)^{j_n}$$ Wouldn't then the minimal polynomial be exactly: ...
0
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1answer
13 views

Prove that the direct sum of a symmetric and skew symmetric matrix belongs to $M_n(K)$ using $A_{ij}$ and $A_{ji}$ notation.

Basically Let $M_n(K)$ be an $n\times n$ matrix of a $K$ vector space. $U =\{A\in M_n(K)\;|\;A_{ij}=A_{ji}\}$ $W =\{A\in M_n(K)\;|\;A_{ij}=−A_{ji}\}$ So I don't understand my mark scheme. It says ...
0
votes
1answer
340 views

Find an ordered basis $B$ for Mnxn(R) such that [T]$B$ is a diagonal matrix for n > 2?

I have a homework problem that I'm stuck on. It is problem 5.1.17 in the Friedberg, Insel, and Spence Linear Algebra book for reference. "Let T be the linear operator on Mnxn(R) defined by $T(A) = ...
3
votes
1answer
39 views

Double dot product in Cylindrical Polar coordinates - Strain energy

I'm working with a problem in linear elasticity, and I have to calculate the strain energy function as follows: $$ 2W = σ_{ij}ε_{ij} $$ Where σ and ε are symmetric rank 2 tensors. For cartesian ...
0
votes
1answer
41 views

Shortest distance between parallel line and plane

I've been doing questions regarding the shortest distance between lines/planes and points , and I've come across a question asking to find the shortest distance between a line and a plane which are ...
0
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1answer
35 views

First-order linear differential equation for matrix valued functions of size $3\times 3$

I have two matries given by (M' means derivative w.r.t x) $ M=\left( \begin{array}{ccc} f_1(x) & f_2(x) & f_3(x) \\ f_4(x) & f_5(x)& f_6(x) \\ f_7(x) & f_8(x) & ...
0
votes
1answer
34 views

Prove that $\exists$ U: $P$ is self adjoint if and only if $P=P_U$

Suppose $P \in L(V)$ is such that $P^2 = P$. Prove that there is a subspace U of V such that $P= P_U$ if and only if P is self adjoint. First suppose that $P = P_U$ Show this implies that P is self ...
1
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0answers
23 views

How to see that $\text{dim}(L)=k-1$?

Consider $L:=\left\{x\in\mathbb{R}^k: cx=\delta\right\}$ with $c=(c_1,\ldots,c_k)$ and $\delta\in\mathbb{R}$. Show that $\text{dim}(L)=k-1$. Do not know how to show that. Anyhow my first ...
0
votes
1answer
56 views

$A \cdot B = A \cdot C$ does not imply that $B = C$

I am trying to prove the following: If $A,B,C$ are non-zero vectors such that $A \cdot B = A \cdot C$, then it's not necessarily true that $B = C$. My proposed proof. Suppose $A \cdot B = ...
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votes
2answers
75 views

Prove that $T-\sqrt{2}I$ is invertible.

Suppose $T \in L(v)$ is such that $\|Tv\| \leq \|v\|, \forall v \in V$. Prove that $T-\sqrt{2}I$ is invertible. I know that I need to show there exists a $R^{-1}(T-\sqrt{2}I) = I$ where $R = ...
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2answers
50 views

Solving Systems of Equations..!! [on hold]

I actually don't want to solve the systems of equations. I just want to let them equal each other so I can cancel one of them and solve the other one. Example: $E_1\quad x-y=0$ $E_2\quad 2x-2y=0$ ...
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0answers
166 views
+500

vector spaces whose algebra of endomorphisms is generated by its idempotents

Let $V$ be a $K$-vector space whose algebra of endomorphisms is generated (as a $K$-algebra) by its idempotents. Is $V$ necessarily finite dimensional? EDIT (Jul 26 '14) A closely related question: ...
1
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3answers
92 views

Solution of a recurrence with varying coefficient

Please help me to find the solution of the recurrence in terms of n(implies $(f(n))$ and also the summation of the recurrence up to infinity ($sum = \sum_{n=0}^\infty f(n)$) . ...
1
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1answer
43 views

sign determinant $2\times 2$

I have been reading internet and tried to understand the explanation of the sign of a determinant of a $2\times 2$ matrix. if I have a matrix \begin{array}{cc} a & b \\ c & d\\ ...
21
votes
10answers
12k views

How to show that $\det(AB) =\det(A)\det(B)$

Given two square matrices $A$ and $B$, how do you show that $\det(AB) = \det(A)\det(B)$, where $\det(\cdot)$ is the determinant of the matrix?
0
votes
2answers
597 views

Volume of a Pyramid Linear Algebra

Find the volume of a pyramid with triangular base bounded by vectors (1,-1, 2) and (1, 1, 1) and vertex located at (3, 2, 5). I am not sure how I would solve this. I know the volume of a pyramid is: ...
1
vote
2answers
38 views

Show that 2 matrices belong to a square matrix by taking the transpose. Vector spaces

Let $M_n(K)$ be an $n\times n$ matrix of a K vector space. \begin{align} U &= \{A ∈ M_n(K) | A_{ij} = A_{ji} \} \\ W &= \{A ∈ M_n(K) | A_{ij} = -A_{ji} \} \end{align} Prove that $U$ and $W$ ...
1
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1answer
217 views

Subspaces, transformation matrices exercise

I have trouble understanding the following exercise so I would really appreciate any help you could give me: Let $k$ be a non zero vector in $\mathbb R^n$, written in standard basis. Let $H$ be ...
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0answers
35 views

Sparsity of Linear Diophantine Equations

If you are looking for integer solutions to the system. $$Ax=b$$ where $A$ is an integer matrix and $b$ is integer vector, then you can construct the solution space integer matrix $B$ and integer ...
2
votes
1answer
47 views

Is it true that for an inner product space a norm of a vector is defined unambiguously?

Suppose we have some vector space $V$ for which we have defined an inner product $\langle \cdot\rangle$. Thus we have an inner product space. Is it true that $\forall x \in V : \lVert x\rVert := ...
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votes
2answers
52 views

How to find this Linear Transformation

Q. Find the Linear Transformation $T:V_3\rightarrow V_3$ , such that $T(0,1,2)=(3,1,2)$ $T(1,1,1)=(2,2,2)$ I tried considering $(0,1,2),(1,1,1)$ as basis, it doesnt seem to work that way. Just need ...
1
vote
2answers
45 views

the rank of a linear transformation

Let $V$ be vector space consisting of all continuous real-valued functions defined on the closed interval $[0,1]$ (over the field of real numbers) and $T$ be linear transformation from $V$ to $V$ ...
5
votes
2answers
1k views

Is there a unique solution for this quadratic matrix equation?

The quadratic matrix equation I've been looking at lately: $$ Q_{r,r}=A_{r,r}X_{r,r}^2+B_{r,r}X_{r,r}+C_{r,r}=0_{r,r} $$ Note that $A, B, C,$ and $X$ are $r \times r$ matrices. $A$ contains known ...
0
votes
2answers
18 views

Matrix representation of a linear operator

As I'm studying for my final, my book keeps skipping alot of steps and I don't know how tthey get from point a to point b - probably because its elementary at that stage in the book, except not to me ...
0
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0answers
12 views

How to decompose the given matrix by Geometric mean decomposition??

$$H=\begin{pmatrix}1&2&3\\4&5&6\\7&8&9\end{pmatrix}.$$ How to decompose this matrix using Geometric mean decomposition?? If anybody knows this method kindly update how to ...
0
votes
0answers
17 views

Recursion on Matrix

We have a given matrix recurrence given, $ (\curlyvee_i,\curlyvee_{i-1})_{1\times3}= (\curlyvee_{i-2},\curlyvee_{i-3})_{1\times3}{\begin{bmatrix}A_{i-1}A_i+B_i & A_{i-1} \\B_{i-1}A_i & ...
0
votes
2answers
31 views

Solve nonliner equations

We are trying to find intersection of hyperbolas and we ended up in five equations $$\begin{align} A_1X^2+B_1Y^2+C_1XY+D_1X+E_1Y+F1&=0\\ A_2X^2+B_2Y^2+C_2XY+D_2X+E_2Y+F2&=0\\ ...
2
votes
2answers
33 views

$m \times n$ matrix where $m < n$

So I'm a long distance student and I need some help to bounce ideas off of other people who understand the work. Fellow students are few and far between. So while this is an assignment question, I ...
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1answer
15 views

Dimension of $\cal V^-$ if $\cal V$ is a $\Bbb C$ vector space

In Halmo's book 'Finite dimensional vector spaces' there's a question I'm kind of stuck on in chapter 1. $1 (b)$ Every complex vector space $\cal V$ is intimately associated with a real vector space ...
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2answers
33 views

Prove another matrix is positive definite given that A is a Hermitian matrix

Suppose that $A$ is a Hermitian symmetric $n\times n$ matrix of complex numbers all of whose eigenvalues lie inside the interval $(-1,1).$ Prove that the matrix $A^3+Id$ is positive definite. An ...
2
votes
1answer
18 views

Linear Mapping in a Graph

Let $T$ be a linear mapping and $G$ the set of points limited by the triangle $abc$. Find $T$ and represent the image of the graph $G$ through $T$. I have not idea how to find the matrix $T$, and ...
0
votes
1answer
17 views

Spanning sets and Linear Transformations

Suppose $v_1, \dots v_n$ spans V and $T \in L(V, W)$. Prove that the list $Tv_1, \dots , Tv_n$ spans rangeT. I said that if $v \in V = a_1v_1 + \dots + a_nv_n$ then $T(a_1v_1 + \dots + a_nv_n) = ...
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1answer
40 views

Representing translation by matrix multiplication in higher dimension

Problem There is a translation (shift) by vector $t$. If we want to display this shift as a matrix multiplication by T, what are the dimensions of T (number of rows and columns)? Progress I think ...
1
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2answers
30 views

Base of Subspace with vectors

Let E be the vector subspace of $R^3$ generated by it vectors $v1 = (1,2,0)$ and $v2 = (-1,0,2)$ How can find a basis of E between the following vectors? $$w1=(-2,-12,8), w2=(-12,-2,-8), ...
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0answers
33 views

Proof that $S^\perp$ is a subspace of a vector space $V$

Just doing some review for a final exam and would like some feed back on the following proof if anyone would like to help me out. First the premise. Let $V$ be a finite dimensional inner product ...
12
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6answers
7k views

Matrix Inverses and Eigenvalues

I was working on this problem here below, but seem to not know a precise or clean way to show the proof to this question below. I had about a few ways of doing it, but the statements/operations were ...
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2answers
23 views

Line parallel to a plane and have 45 degrees between another

I need to find a direction vector for a line parallel to a plane $x+y+z = 0$ and that have $45$ degrees with the plane $x-y = 0$ So, i've assumed the vector $\vec V_r = (a,b,c)$ and since it is ...
1
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6answers
114 views

How to raise a matrix to the power of $13$ without boring, repetitive multiplication?

how can i show $\begin{pmatrix}1 & 1 & 1 \\0 & 1 & 1 \\0 & 0 & 1\end{pmatrix}^{13}=\begin{pmatrix}1 & 13 & 91 \\0 & 1 & 13 \\0 & 0 & ...
6
votes
1answer
40 views

Prove that $v_1, \dots v_n$ is a basis of V.

Prove that if $e_1, \dots e_n$ is an orthonormal basis of V and $v_1, \dots , v_n$ are vectors in V such that $||e_j - v_j|| < \frac{1}{\sqrt{n}}$ for each j, then $v_1, \dots v_n$ is a basis of ...
1
vote
1answer
76 views

If $v_1, \dots, v_m$ are linearly independent, then there is $w$ such that $\langle w, v_j \rangle > 0$ for all $j$

Suppose $v_1, \dots v_m$ is a linearly independent list in V. Show that there exists $w \in V$ such that $\langle w, v_j \rangle > 0$ for all $j \in {1, \dots ,m}$. I understand this question is ...
0
votes
1answer
40 views

Characterization of a matrix with eigenvalues equal to one

Consider an $m\times m$ non-negative matrix $A$ where elements of $A$ can take many different values e.g. they are functions of a variable z. Suppose $A$ is such that one of its eigenvalues is equal ...