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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19 views

Prove that $(f, g, h)$ is a linearly independent list of vectors in $\mathbb{R}[x]^S$

"Recall that $\mathbb{R}[x]$ is a vector space. Suppose that $f, g, h \in \mathbb{R}[x]^S$ and that there is $q \in S$ such that $f(q) = 1$, $g(q) = x^2 + 1$, and $h(q) = x^2 + x$. Prove that $(f, g, ...
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1answer
24 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 ...
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0answers
21 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
62 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 ...
-1
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3answers
27 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
67 views

An excercise problem on Hoffman Linear Algebra

Let $V$ be the vector space over $\mathbb R$ of all functions $f :\mathbb R \to\mathbb R$, then identify if the following is a subspace of $V$: All $f \in V$ such that $f(x^2)=f(x)^2$ While I ...
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1answer
38 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 ...
3
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3answers
160 views

determinant of the linear transformation $T(X) =\frac{1}{2} (AX+XA)$

Let $V$ vector space of all matrices $3\times3$, and let $A$ be the diagonal matrix : $$ \begin{pmatrix} 1 & 0 & 0\\ 0 & 2& 0 \\ 0 & 0& 1\end{pmatrix} $$ Compute thee ...
20
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4answers
14k views

How to find basis for intersection of two vector spaces

What is the general way of finding the basis for intersection of two vector spaces? Suppose I'm given the bases of two vector spaces U and W: $$ \mathrm{Base}(U)= \left\{ \left(1,1,0,-1\right), \left(...
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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
60 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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2answers
26 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,...
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 &...
0
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1answer
33 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 ...
21
votes
2answers
2k views

Cardinality of a Hamel basis of $\ell_1(\mathbb{R})$

What is the cardinality of a Hamel basis of $\ell_1(\mathbb R)$? Is it deducible in ZFC that it is seemingly continuum? Does it follow from this that each Banach space of density $\leqslant 2^{\...
0
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4answers
62 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 $0$ ...
6
votes
4answers
620 views

Is $\mathbb{C}^2$ isomorphic to $\mathbb{R}^4$?

Are $\mathbb{C}^2$ and $\mathbb{R}^4$ isomorphic to one another? Two vector spaces are isomorphic if and only if there exists a bijection between the two. We can define the linear map $T: \mathbb{C}^2 ...
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
37 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 ...
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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
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 ...
1
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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 $\langle L|p(x)\rangle$ ...
0
votes
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 ...
0
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0answers
30 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 $...
1
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1answer
36 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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3answers
42 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 "...
0
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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
33 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
66 views

Natural Transformaton $\text{Hom}(V,W)$ and $W\otimes V^*$

Something of this form has already been answered here: Why is $\text{Hom}(V,W)$ the same thing as $V^* \otimes W$? I'm starting introductory category theory stuff, and I'm looking for some help. I ...
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0answers
26 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
50 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 (...
1
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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 ...
0
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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 ...
2
votes
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 ...
0
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1answer
34 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 +...
0
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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\...
0
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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
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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2answers
59 views

If I have a vector inside a rectangle, how do I tell which side of the rectangle the vector will hit?

I'm trying to solve an issue where I basically have a vector inside of a rectangle. I want to figure out if the vector continues its trajectory, what side will it strike? The vector is given as an (x, ...
0
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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-...
2
votes
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 ...
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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
votes
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 ....
2
votes
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
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 ...
1
vote
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
votes
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 ...