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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18 views

Is it better to average the log2 for a series of numbers or just the numbers themselves? And, how would you test or prove this?

Lets say I'm trying to compare two vectors for similarity and normalizing them before hand based on some mean or standard deviation combo for the purpose of finding the similarity between the 2 ...
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1answer
58 views

Differentiation w.r.t. the $\mbox{vec}$ operator

I am stuck at solving the following derivative $$\frac{d \mbox{vec} (X^T X)}{d \mbox{vec} (X)}$$ where $X$ is an $m \times n$ matrix and $\mbox{vec}$ is the vector/stack operator. I have tried using ...
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3answers
26 views

Unit vector c perpendicular [closed]

Find a unit vector $c$ perpendicular to both of the vectors $a = 0j + 1j - k$ and $b = 2i + 2j – k$. Just need steps/hints or even the solution would help me check if I go it right.
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1answer
37 views

Construct a linear map $M : V → V$ with the property that $K = \{v ∈ V\mid Mv = 0\}.$

"Suppose that V is a vector space and $L : V → V$ is a linear map. (i) Let K ⊂ V be the set of all vectors $v ∈ V$ such that $L(v) = −v$. Show that K is a subspace of V . (ii) Construct a linear ...
2
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1answer
31 views

Clarification of ideas concerning a quotient space.

Suppose I have a vector space $V$, and I identify $x\in V$ with $\lambda x\in V$, where $x\neq 0$ and $\lambda>0$, $\lambda\in\mathbb{R}$. I'm confused about two things: (1) Can I define a norm on ...
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1answer
20 views

Definition of complex vector space from Rudin RCA

This is definition of complex vector space from Rudin's book. He write that to each pair $(\alpha,x)$, where $x\in V$ and $\alpha$ is scalar there is associated a vector $\alpha x\in V$. That's right. ...
2
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3answers
59 views

Why a linear trasformation doesn't depend on the bases we choose

Imagine we are given the following linear transformation: $f(x,y) = (x+y, x)$ Imagine we choose a base, let's call it $B_{1}$ and we apply the function to some vector. Now imagine we choose another ...
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0answers
14 views

Show that V1 is a linear subspace of R[x]?

"Let $\mathbb{R}$ be the set of polynomials, and let $ V_1 = (a_1x + a_2x^3 + a_3x^5$ | $a_1, a_2, a_3 \in \mathbb{R}$ ) and $ V_2 = (b_1x^2 + b_2x^3 + b_3x^4$ | $b_1, b_2, b_3 \in \mathbb{R}$ ) be ...
1
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1answer
21 views

Understanding components of a vector

I learned that we can get the component of a vector in any direction using the dot product. The problem I have is the meaning of the term component itself. The component of a vector $\vec A$ in the ...
1
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2answers
25 views

Show that $V = \ker T \oplus \operatorname{im}T$ where $T$ is an idempotent linear operator [duplicate]

I have to prove that if $T$ is an idempotent ($T^2=T$) linear operator then space $V = \ker T\oplus\operatorname{im}T$. My first try was to think about the basis of subspace $\ker T$. Let say $(e_1,...
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1answer
31 views

Non-negative Linear Span of Vectors

I would like to understand if there is a common concept of a `linear span' of a set of vectors which are combined with non-negative multipliers. I know that usual definition of the span of a set of ...
0
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4answers
57 views

How to prove a $W=\{(x,y):3x+y=0\}$ is a subspace of $ R^2$

How can I prove this vector $W$ is a subspace of $\mathbb{R}^2$ (closed under addition and scalar multiplication) if I have the condition $3x+y=0$. Does this mean this vector already has the $0$ ...
0
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4answers
50 views

Direct sum of vector subspaces equals $\mathbb R^3$

I tried solving the following linear algebra problem, I hope that someone can tell me if this is a good solution, and if not, how I should solve it. Let $U$ and $W$ be vector subspaces of the vector ...
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0answers
36 views

How to fill the basis of vector subspace up to $\mathbb{R}^3$?

So, if we're given a vector subspace $V$ of $\mathbb{R}^3$ with basis: $$B_V=\{(-1,1,1),(2,1,-1)\}$$ How can we find a basis of vector subspace $F$ such that: $$V\oplus F=\mathbb{R}^3 \ ?$$ What I did ...
1
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1answer
26 views

what is the dimension?

The dimension of the row space of a $8\times 8$ matrix $A$ is 5.if $\mathbb{R}^{8\times 10}$ is a vector space of $8 \times 10$ matrices with real entries. Then $S_{A} = \{ B \in \mathbb{R}^{8\times ...
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0answers
43 views

Action of a Linear Functional on a Polynomial

I was hoping to find a good canonical reference for the mathematics behind something called the action of a linear functional $L$ on a polynomial $p(x)$ which is denoted $\langle L|p(x)\rangle$ ...
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0answers
36 views

What operator has these algebraic properties?

I am working in a space $V$ of objects that behaves like a vector space with a partial ordering $\preceq$. I have discovered an operator $f:V\times V \rightarrow V$ with the following properties: For ...
0
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2answers
22 views

Subspace proof wording

not sure how to word my answer for this question: Let V be a vector space and let H and K be two subspaces of V. Show that the following set W is a subspace of V: W={u+v: u ∈ H, v ∈ K} I'm pretty ...
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0answers
29 views

Linear Algebra Axler, 3e, Exercise 1c, P11

P11. Prove that the intersection of every collection of subspaces of V is a subspace of V. My solution was very similar to the one from linearalgebras.com -- Solution: Assume $U_i$ are subspaces of $...
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0answers
17 views

how do project a point to a line?

I have a situation like this enter image description here I know the point n and the point s, and the distance between those two, and the tangent of the point n. I need to somehow project the ...
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1answer
35 views

What's the difference between $V\times W$ and $V\otimes W$ where $V$ and $W$ are vector space?

Let $V,W$ vector spaces. I don't really understand what is $V\otimes W$. To me it looks the same that $V\times W$. Do you have any explanation ?
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0answers
32 views

Canonical orientation on a complex vector space

I know that this is much of a favor I'm asking you for, but since this specific task is due by tomorrow and since we are really fighting for our admission, I wanted to ask you if you have a link to a ...
1
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3answers
40 views

Set of orthogonal vectors in $\mathbb{R}^n$

How can we show that a set of pairwise orthogonal vectors in $\mathbb{R}^n$ has size at most $n$? I know it seems very intuitive, but not sure what the formal proof would look like (whether "...
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0answers
21 views

when can i move a sum through this tensor product?

If I have a vector space $V^{(1)}\otimes V^{(2)}$ and I have some ray $\sum\limits_k x_k s_k\otimes s'_k = s\otimes \sum\limits_k x_k s'_k$, is the only solution that $s_k = s$ $\forall$ $k$? All $x_k$...
0
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2answers
32 views

vector and curl identity

This popped up in my notes and the author made no remarks about the properties used $\bigtriangledown \times \left ( \vec{E}+\frac{\partial \vec{A}}{\partial t} \right )=\vec{0}$ Then, $\...
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0answers
24 views

Prob. 4, Sec. 4.3 in Kreyszig's functional fnalysis text: Application of the Hahn Banach Theorem

Let $X$ be a real or complex vector space, and let $p \colon X \to \mathbb{R}$ be a real-valued function satisfying $$p(x+y) \leq p(x) + p(y) \ \mbox{ for all } \ x, y \in X$$ and $$p(\alpha x) = \...
3
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1answer
49 views

Dimension of set of $3\times 3$ matrices?

Calculate the dimension of the image and kernel of each linear transformation. (Hint: you do not need to find a matrix representing the linear transformation.) $(a)$ $P\colon\Bbb R^3\to\Bbb R^3$ by (...
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1answer
52 views

Subspace proof $\{\,f \colon\Bbb R \to\Bbb R \mid f(x + 1) = f(x) + 1\,\}$

I have no idea how to show that this is a subspace. Isn't $f(x)=x$ and $f(x)=3x$ a counter-example? It is not closed under scalar multiplication? But I guess it is.. $[e]$ forgot to say that the ...
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1answer
51 views

A question about diagonalizable linear operator

Suppose $T$ is a diagonalizable linear operator on a finite dimensional vector space $V$. Prove $V$ is T-cyclic subspace of itself iff every characteristic subspace of it is one-dimensional. It ...
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1answer
33 views

switching between P3 and R4 vector spaces

first of all I couldn't find a better way of describing what I really meant so here it goes. Lets say I have a vector space $\mathbb{P}_3[\mathbb{R}]$ and a sub vector space $U = \textrm{span}\{ x^2 +...
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0answers
21 views

Hermite Polynomials using gram-schimdt

How does one generate these polynomials using the gram-Schmidt algorithm? I know how it should work, but I get 0 as the value for the scalar product of (p1,q0) and q1 should be 2x not x. $$q1\left(x\...
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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
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2answers
40 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
21 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-...
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1answer
43 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
votes
3answers
50 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 ....
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2answers
73 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
87 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}...
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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]...
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1answer
44 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
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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 ...
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0answers
21 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
32 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 $\...
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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
18 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
30 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
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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 ...