For questions about vector spaces and their properties. More general questions about linear algebra belong under the [tag:linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars. In other words, these are the spaces ...

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2answers
33 views

Plotting a 3D graph from explicit equation

I´m a 2nd year engineering student and today we learned how to plot 3d graphs from a $XYZ$ equation on paper. For example, I know ($\frac{X^2}{9}+ \frac{Y^2}{16} + \frac{Z^2}{9} =1$) will produce an ...
2
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1answer
75 views

Sandwich rule for Lie algebras

On an infinite dimensional vector space an operator can be onto but not one-to-one (and vice versa). This arises the following question. Let $L_1$ and $L_2$ be Lie algebras (infinite dimensional, over ...
2
votes
2answers
44 views

Endomorphism is normal and idempotent iff it is an orthogonal projection.

I've searched for answers for this question here for some time but haven't found an applicable answer because I could only find related questions, but not this one in particular. Suppose $V$ is a ...
2
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0answers
38 views

How do I compute the gradient of a tensor?

From this paper, we have three matrices $U\in \mathbb{R}^{n\times d_U}$, $M\in \mathbb{R}^{m\times d_m}$, $C\in \mathbb{R}^{c\times d_C}$ and a tensor $S\in \mathbb{R}^{d_U \times d_M \times d_C}$, ...
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0answers
22 views

Is there a name for a $k$-sphere embedded in $\mathbb{R}^n$?

In my thesis in a lot of places there comes up a $k$-sphere embedded in $\mathbb{R}^n$. We call lower-dimensional "planes" in $\mathbb{R}^n$ linear manifolds or flats, is there also a term for a lower-...
1
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1answer
46 views

How to find a basis for subspace of functions

I am doing this exercise: The cosine space $F_3$ contains all combinations $y(x) = A \cos x + B \cos 2x + C \cos 3x$. Find a basis for the subspace that has $y(0) = 0$. I am unsure on how to ...
2
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3answers
51 views

Rule for $\langle x,y\rangle$ if we know orthonormal base?

How to define $\langle x,y \rangle$ in space of polinoms, where $1, x-1 , 1-x^2$ are orthonormal base($\Vert a\Vert = 1$, $\langle a1, a2\rangle = 0$)? I'm a bit lost, I know how to do it with my ...
2
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3answers
27 views

Question about Column space and Row space of generic matrix $A$ and the corresponding upper triangular $U$

I am doing the following exercise from Introduction to Linear Algebra: Find the dimensions of (a) the column space of A, (b) the column space of U , (c) the row space of A, (d) the row space of U ....
1
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2answers
78 views

A possible generalized determinant?

This will likely seem a bit contrived, and admittedly it is, but I wanted to see just how "close" we could get to generalizing the concept of a determinant. In what follows, we will lose quite a few ...
2
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2answers
88 views

$2\times 2$ real matrix with exactly one eigenvalue [duplicate]

Problem: Let $A$ be a $2\times 2$ real matrix with exactly one eigenvalue $\lambda \in \mathbb{R}$, but that $A \not= \lambda I $, show that there exists an invertible matrix $P$ such that $$ P^{-1}...
1
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1answer
27 views

Are the norms on a vector space unique?

I was watching an online lecture on bounded linear transformations $$T: \mathcal{C}[a,b] \rightarrow \mathcal{C}[a,b]$$ So the condition for $T$ to be bounded was that for all $f \in \mathcal{C}[a,b]...
0
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1answer
45 views

why bother with extra orthonormal vector in Singular value decomposition

when we do the SVD for a $m\times n$ matrix, we have to extend the set $u_1, ... , u_r$, to an orthonormal basis $u_1, ... , u_m$ for $R^m$ if $r<m$. But why don't we just fill zero vectors to make ...
1
vote
1answer
20 views

Vectors in Normed Space Must Have Finite Length?

I have assumed this to be the case, and consequently this is why one looks at convergent sequences of vectors in normed, Banach, and Hilbert spaces. But, I've never seen this listed explicitly as an ...
0
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0answers
22 views

Fourier basis of $L^2([-\pi,\pi])$

I have read that the Hilbert space $L^2([-\pi,\pi])$ has a Hilbert basis: $$\{e^{inx}|n\in\Bbb{Z}\}$$ This to me indicates that we can only represent a function $u(x)$ by a Fourier Series iff $u(x)\in ...
0
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2answers
35 views

Generalization of inner product

I was wondering if there was a widely accepted generalization of inner product spaces where the inner product look something like $\langle\bullet , \bullet\rangle:V\times V \to \mathbb{F}$, where $\...
1
vote
1answer
29 views

How to solve this linear algebra problem(Space of diagonal matrices)?

We have space M of 3x3 matrices. Our scalar product is defined as = tr(AB^t) a) We have vector sub space D of diagonal matrices. Find base and dimension of orthogonal complement of D. Any hint ...
0
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1answer
20 views

Tensor product space: dual of the space of bilinear functionals on the Cartesian product

My reading (link provided for completeness only, clicking is not necessary) defines the tensor product space as follows: Let $V$ and $W$ be vector spaces. The symbol $v\otimes w$ is defined to be the ...
0
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1answer
32 views

Why do we study specifically 'normed' vector spaces?

When we study vector spaces, it is useful to define a norm on it for countless reasons. I was thinking about this recently and realised Don't all vector spaces have norms on them? If they all ...
1
vote
1answer
42 views

Constructing Vector Spaces from Linear Combinations

I've modified parts of this quote from "Introduction to Linear Algebra by Strang", for brevity in the questions below. If the quote itself seems illogical or incorrect in any form, please inform me ...
0
votes
1answer
26 views

Distance between point and parametric line

Compute the distance between the point R(1, 1, 1) and the line (0, -1, 1) + t(1, -2, 2) Can someone please show me how to answer this in full? It was in a test recently and is still bugging me as the ...
1
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2answers
51 views

Cauchy_Schwarz Inequality

When does equality hold in the Cauchy-Schwarz inequality |x • y| ≤ ||x|| · ||y|| for vectors x, y ∈ R^n? Can someone please show me how to answer this in full? It was in a test a few weeks ago and ...
0
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0answers
27 views

Midpoints of a parallelogram [duplicate]

In the parallelogram ABCD the point E is the midpoint of AB, and F is the midpoint of BC. Show that the line segments DE and DF divide the diagonal AC into three equal length pieces. Can someone ...
0
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1answer
34 views

How to find the value of A and B

The value of dot product and the cross product of two vectors $\vec{A}$ and $\vec{B}$ are given as $\sqrt{3}$ and $ 2 \sqrt{3}$. I have to find out the value of A and B. I can easily find the ...
0
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1answer
49 views

Divergence theorem to Stokes' theorem, is it possible?

Is it possible to show that Divergence theorem is more fundamental? Can we arrive at Stokes' theorem from Divergence theorem?
2
votes
1answer
26 views

Linear independence of functionals imply null coeficients on sum?

Let $V$ be a vector space over a field $K$, $V^*$ be it's dual (it's linear functionals), $\{\alpha_1,...,\alpha_n\}$ be a basis for $V$ and $\{f_1,...,f_n\}$ be the dual basis. Any subset $S^*$ of ...
0
votes
1answer
33 views

Does the following function define a distance metric?

For real numeric vectors of length $N$, let $a_n \succ b_n$ be one if true and zero if false. The distance between $A$ and $B$ is $$\sum_1^N a_n \succ b_n$$ Note that this is very similar to the ...
0
votes
0answers
25 views

Find the orthonormal projection?

I am trying to find the orthonormal projectiont of $X\in V\equiv M_{2\times2}(\mathbb{R})$ over the space of diagonal matrix defined by $$D=\{M=(m_{ij})\in V:m_{ij}=0,\ i\not=j\}\ \ \ i,j\in\{1,2\}$$ ...
0
votes
2answers
64 views

Relation between the dual space, transpose matrices and rank-nullity theorem

Summing up, how can one use linear functionals, transpose matrices, row and column rank equality and annihilators to prove the rank-nullity theorem? While studying linear algebra, I'm trying to get ...
9
votes
1answer
165 views

Invariant vectors of $A^n B^m$ with $A,B$ orthogonal matrices

Let $A$ be the following matrix:$$A=\dfrac{1}{2}\ \left( \begin{array}{cccccccccc} -1 & -1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & 1 & 1 & 1 & 0 &...
2
votes
0answers
26 views

Transformation matrix is jordan normal form

I have the following question: Given a finite-dimensional, unitary vector space V and a endomorphism f on V, is it possible to choose an orthonormal basis B of V in such a way, that the transformation ...
0
votes
2answers
76 views

Is span $\{[1,0],[0,1]\}$ a vector space?

I can't figure this out. I would think that it is a vector space because it has the zero vector, and it seems to me to be closed under addition and scalar multiplication. But $[1,0]+[0,1] = [1,1]$ ...
2
votes
3answers
24 views

Show that $f(y)=0$ for every $y \in Y$.

If $Y$ is a subspace of a vector space $X$ and $f$ is a linear functional on $X$ such that $f(Y)$ is not the whole scalar field of $X$, show that $f(y)=0$ for all $y \in Y$. Suppose to the contrary, ...
0
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0answers
19 views

If $U_0,V$ are Hilbert spaces, $(e_n)$ is an ONB of $U_0$ and $ι:U_0→V$ is an embedding, can we complete $(ιe_n)$ to an ONB of $V$?

Let $U$ and $H$ be Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric $U_0:=Q^{1/2}U$ be equipped with $$\langle u,v\rangle_{U_0}:=\langle Q^{-1/2}u,Q^{-1/2}v\rangle_U\;\;\;\text{for }...
0
votes
0answers
39 views

Alignment of one 3D Coordinate system to another 3D Coordinate system

I'm working on a project depicted by this picture(taken from internet) where there are different coordinate system involved which corresponds to camera coordinate system and local 3D coordinate system ...
0
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0answers
27 views

Proof Verification: Separable Hilbert Space has a Countable Orthonormal Basis?

I was browsing proofs of this involving the Gram-Schmidt process when the following occurred to me. I'd appreciate feedback. If $H$ is a separable Hilbert space then it has a countable dense subset $...
0
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0answers
20 views

Relating $\mathrm{dim}_\mathbb{R}$ of a complex vector space to the $\mathrm{dim}_\mathbb{R}$ of its decomplexification

Let $V$ be a vector space over $\mathbb{C}$. By restricting scalar multiplication to $\mathbb{R}$, we obtain another vector space, say $V_\mathbb{R}$, over $\mathbb{R}$; from what I understand, this ...
1
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1answer
12 views

Any minimal spanning set must be the maximal independent set?

By definition: A minimal spanning set is a spanning set such that no proper subset thereof is spanning. A maximal independent set is an independent set such that no set that contains it properly ...
1
vote
1answer
18 views

Basis of $\text{sl}(3,\mathbb C)$

I would like to have a basis of $sl(3,\mathbb C)$, the 3 times 3 matrices with complex entries such that their trace is zero. I know that this vector space is 8-dimensional but I struggle in finding ...
0
votes
1answer
32 views

Multipart Linear Algebra Problem

Hi I need a bit of help on this Linear Algebra problem that was on past test in my LA course. Let T be a operator on the space R3 defined with: T((x,y,z)) = (x + 2y - z, y + x, -x +2y + 4z). a) ...
0
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0answers
24 views

warping a cube in a 3d space

3D cube made with Octave The problem I have is shown in the picture above. I have 2 cubes where both cubes are filled with vectors, or have possible vector locations. The blue cube is filled normally,...
1
vote
1answer
27 views

Vector space $V\cong \mathbb R^n$ or $\cong \mathbb C^n$.

I was wondering something, Let $V$ a $\mathbb R-$vector space and $W$ a $\mathbb C-$vector space both of dimension $n$. Is it true that $$V\cong \mathbb R^n\quad \text{and}\quad W\cong \mathbb C^n\ \ ...
1
vote
2answers
35 views

Set of all $2*2$ matrix on $\mathbb Q$ is not a vector space over $\mathbb R$

why $ V= \{\begin{pmatrix} a & b \\ c & d \\ \end{pmatrix}|a,b,c,d \in \Bbb Q\} $ is not a vector space over $\Bbb R$ under usual matrix addition and scalar ...
5
votes
3answers
55 views

Is it okay to think of functions as of vectors with “uncountable index”

In some applied areas that have a little scent of functional analysis (e.g., getting error bound in numerical methods), it is somewhat appealing for me to think of functions $\mathbb R \to \mathbb R$ ...
1
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1answer
20 views

Why is left-inverse $A_L^{-1}$ times any column vector $b$ in row space of $A$?

Let $A$ be a matrix with a left inverse $A_L^{-1}$. That is, $A_L^{-1} = (A^TA)^{-1}A^T$. Then for any column vector (with proper dimension) $b$, $A_L^{-1} b$ is in the row space of $A$. i.e., $A_L^{-...
5
votes
1answer
41 views

Prove that there is a base of $\mathbb R^4$ made of eigenvectors of matrix $A$

Matrix of linear operator $\mathcal A$:$\mathbb R^4$ $\rightarrow$ $\mathbb R^4$ is $$A= \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1\\ 1 &...
0
votes
1answer
27 views

Confusion with some theorems leading to the canonical decomposition of an operator

Let $\Bbb K$ be a field and $R = \Bbb K[X]$ be the ring of polynomials with coefficients in $\Bbb K$. Let $\cal L_{\Bbb K}$$(V)$ denote, for a finite dimensional $\Bbb K$-vector space $V$, the set of ...
1
vote
2answers
37 views

What does the sum of subsets of a vector space mean?

On page $57$ of Second edition of Hoffman Kunze, the authors write Definition If $S_1, S_2, \dots, S_k$ are subsets of a vector space $V$, the set of all sums $$\alpha_1 + \alpha_2 + \dots+ \...
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0answers
20 views

How can I prove algebraic and topological dual spaces do not coincide in infinite dimensional normed vector spaces?

I've heard it's enough to give an example of a non continuous linear functional, but I'm kinda confused, because some definitions ask for "bounded" at infinite spaces, does bounded mean continuous in ...
0
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2answers
39 views

Prove/Disprove question on matrix vector multiplication and linear independence [closed]

If $\left\{Bv_1, \ldots , Bv_k\right\}$ is a linearly independent set in $\mathbb{R}^k$ where $B$ is a $k \times n$ matrix, then $\left\{v_1, \ldots ,v_k\right\}$ is a linearly independent set in $\...
0
votes
0answers
16 views

Show $\{\beta (v,w) =0 \Leftrightarrow \beta(w,v)= 0\} \:\:\:\: \Rightarrow \:\:\:\: \{\beta(w,u)\beta(u,v) = \beta (v,u) \beta (u,w) \}$

$\beta$ is a bilinear form on a $K$-vectorspace $V$. Now i have to show the following: if $\forall$ $v,w \in V$ $$\beta (v,w) =0 \Leftrightarrow \beta(w,v)= 0 $$ then $\forall u,v,w \in V$ $$\beta(...