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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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 ...
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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 ...
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 ...
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 ...
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 ...
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 ...
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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 ...
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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 ...
1
vote
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: ...
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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 ...
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\\ ...
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) & ...
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 ...
1
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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 $W$ ...
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 := ...
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 ...
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votes
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 ...
0
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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
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 ...
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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 ...
1
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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 ...
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 ...
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\\ ...
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) = ...
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 ...
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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 ...
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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), ...
0
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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 ...
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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 = ...
1
vote
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 ...
4
votes
1answer
73 views

Why generalize the derivative for multivariable functions? [duplicate]

Sorry if this is a dupe (did a search, couldn't find anything). In single variable calculus, if the following limit exists: $$\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h},$$ then this expression ...
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 ...
2
votes
3answers
45 views

Let $T:V\to W$ be a linear transformation. If $\dim V> \dim W$ then $T$ is not injective. True or false?

I think it's true, i just did this demo, please can you help me if i'm missing something or doing it wrong. Thanks.
1
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1answer
35 views

Proving that $E=F\oplus G$ for two given subspaces of $E = \mathbb R^3$

Suppose that $F ={(x,y,z)\in \mathbb{R}^3 |x−y+z=0}$ and $g=(1,1,1)$ with $G=Vect(g)$ How can I prove that $E=F\oplus G$? I'm wondering how many ways exist to prove that?
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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 & ...
0
votes
1answer
26 views

Matrix which has every vector in the space as an eigenvector is square, diagonal.

I've been trying to solve this linear algebra problem for some time and have gotten stuck. I've been asked to either prove or disprove the following statement: For $V$ an $\mathbb{ R } $ vector ...
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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 ...
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1answer
48 views

A different vector product

I am taking linear algebra, and have learned about the vector dot product and cross product. Is there a vector product defined by : $(a_1, a_2, \dots ,a_n)\times (b_1, b_2, \dots,b_n) = (a_1b_1, ...
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0answers
9 views

Request for information about certain linear transformations of functions on subsets

Suppose I have an infinite set $U$ and let $M$ be the linear subspace of all real-valued functions $\nu$ on $2^U$ such that $\nu(\emptyset) = 0$. Here the sum of two such functions (and the product of ...
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3answers
108 views

How to show that the set of functions $1,x,x^2,x^3…$ is linearly independent? [on hold]

Show that the set of functions $1,x,x^2,x^3...x^n...$ is linearly independent on any interval $[a,b]$. If $$c_1+xc_2+x^2c_3+x^3c_4...=0$$ we should show $$c_i=0,\quad i=1,2, \ldots$$ how could I ...
0
votes
1answer
22 views

Pseudoinverse and orthogonal projection

Given the matrix $A= \begin {pmatrix} 1 & 1 &1 \\ -1 & 1 & 0 \\ 0 & 2 &1 \end{pmatrix}$. (i) Determine the orthogonal projection $p:\mathbb{R}^3 \rightarrow \mathbb{R}^3$ on ...
2
votes
1answer
51 views

Canonical embedding into dual space?

How would one go about proving that there is no embedding of a vector space into it's dual that is independent of a choice of basis? Thanks
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40 views

Conditional identity based proof [on hold]

Suppose $a,b,c$ are non-zero real numbers such that $a+b+c=0$, prove that $$\frac {a^2}{2a^2+bc}+\frac {b^2}{2b^2+ca}+\frac{c^2}{2c^2+ab}=1$$ I tried this and in a lengthy process I got the proof. I ...
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2answers
60 views

I see some contradiction in the definition of orthogonal vectors

Let's look at a well-known definition of orthogonal vectors: Let $V$ be a vector space. Two vectors $x, y \in V$ are orthogonal to each other when the following condition is fulfilled: $$\langle ...
0
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1answer
35 views

If $A \succeq B$ is it true that $B^{-1} \succeq A^{-1}$

If $A$ and $B$ are two positive definite matrices such that $A - B$ is nonnegative definite, is it true that $B^{-1} - A^{-1}$ is positive definite? The doubt came to me when working with confidence ...
2
votes
2answers
59 views

a question about rank of a matrix

Suppose $A$ is a $m\times n$ matrix. Show that $\mbox{rank}\,A=m$ if and only if there exists a $n\times m$ matrix $B$ such that $AB=I_m$. I have proved the case $AB=I_m$ eventuates ...
0
votes
1answer
15 views

Representation of half-space

For any non-zero $\mathbf{y}\in\mathbb{R}^\mathcal{l}$ one half-space through origin is defined by $H_{\mathbf{y}}^{\leq}(\mathbf{0})=\left\{\mathbf{x}\in\mathbb{R}^{\mathcal{l}}:\mathbf{y}\cdot ...
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0answers
14 views

What happens when scaling a rectangle using a pivot point?

With multitouch screens, you can pinch to zoom. When such a gesture is triggered you are supplied with: An x scale factor; A y scale factor; A x pivot point; A y pivot point. When I have a ...
0
votes
2answers
50 views

Proof of the Schwarz's inequality

Let $V$ be a vector space where dot product is defined. Then the following is true: $$\forall x, y\in V \quad \langle x,y\rangle^2 \leq \langle x,x \rangle\langle y,y \rangle$$ Proof: ...
0
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0answers
20 views

The matrix of a linear transformation and the effect on the columns of the identity matrix: uniqueness!

A theorem says: Let $T$ : $R^n \rightarrow R^m$ be a linear transformation. Then there exist a unique matrix A such that $T(\mathbf{x}) = A\mathbf{x} \forall \mathbf{x} \in R^n$. In fact, $A$ is the ...