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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3
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3answers
232 views

Prove that odd polynomials $f(x)$ of degree $\leq 10$ with $f(-1) = 0$ form a vector space.

Let $P(X)$ be the usual vector space of polynomials in $x$ with real coefficients. Let $U$ denote the subset of $P(X)$ consisting of those elements $f(x)$ which have degree less than or equal to ...
1
vote
2answers
128 views

Definition of an angle in a vector space, law of sines

On a lecture in linear algebra we have been given this definition of an angle in vector space with scalar product $\langle , \rangle$: $\cos \alpha=\frac{\langle u,v\rangle}{||u|||v||}$ Throughout ...
0
votes
3answers
58 views

Show that if $u(t)$ is a unit vector for all $t$ then $u(t)$ and $u'(t)$ are orthogonal for all $t$

Show that if $u(t)$ is a unit vector for all $t$ then $u(t)$ and $u'(t)$ are orthogonal for all $t$. So if $u(t)$ is a unit vector it means that $u(t): \mathbb{R} \to \mathbb{R}$ right? I'm not sure ...
0
votes
1answer
42 views

About a finitely generated algebra over a field.

Let $F$ be a field and let $K$ be an associative $F$-algebra which is finitely generated over $F$. Suppose that there exists an element $y ∈ K$ satisfying the condition that for each $v ∈ K$ there ...
12
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4answers
2k views

Why is an infinite dimensional space so different than a finite dimensional one?

In functional analysis there is a big difference between finite- and infinite-dimensional vector spaces. I have found other questions with nice answers here and here. However, I don't grasp the ...
0
votes
2answers
79 views

Find a matrix relative to new bases

Let matrix $T = \left(\begin{array}{cc}1 & 2 \\1 & -1\end{array}\right).$ Let $e_1 = (1,1)^T$ and $e_2 = (3,1)^T$. $T$ is currently relative to the standard basis. If asked to find $T$ ...
0
votes
1answer
102 views

Find matrix of linear transformation relative to new bases

If $T:\Bbb R^3\to \Bbb R^2$ is a linear transformation, and the matrix of $T$ = $\left(\begin{array}{ccc}0 & 1 & 1 \\0 & 1 & -1\end{array}\right)$. If you use the basis $\{i,j,k\}$ ...
4
votes
1answer
53 views

Signature of the norm on $k$-forms.

I wonder about the signature of the inner product on forms on a vector space equipped with a non-degenerate bilinear of signature $(t,s)$. For definiteness, let $(V_{6}, g)$ be an oriented six-...
0
votes
2answers
48 views

Calculate new(x,y) for a line

Suppose if tanks has to rotate its main gun by $30^\circ$ to hit the target, what will be its new $(x,y)$ coordinate or a formula to calculate it as shown in image? If bullet is fired from the main ...
0
votes
0answers
64 views

Do three valued basis vector elements lead to the fastest discrete Fourier transforms?

When sin() and cos() are approximated to 1, 0 and -1 in the basis vectors in a real or discrete Fourier transform the basis vectors have a lot of elements of zero or in common leading to an algorithm ...
0
votes
3answers
32 views

Change of basis into specific values?

Which $a$ value turns the vector $\mathbf{u}=(9, -1, 6)$ into $\mathbf{u'}=(1, 2, 3)$. (From basis $\mathbf{e_1}$, $\mathbf{e_2}$, $\mathbf{e_3}$ to basis $\mathbf{e'_1}$, $\mathbf{e'_2}$, $\mathbf{e'...
1
vote
3answers
75 views

For arbitrary vectors w and u there is a linear operator T such that T(w) = u?

I need to know whether in an arbitrary vector space $V$, given arbitrary non-zero vectors $v,u\in V$, there is a linear operator $T:V\to V$ such that $T(v) = u$. I know that this statement is true if ...
1
vote
2answers
195 views

Show that if $W$ is a finite dimensional $K$-vector space any linear surjective map $f:W\to W$ is bijective.

Show that if $W$ is a finite dimensional $K$-vector space any linear surjective map $f:W\to W$ is bijective. I feel that the rank nullity theorem is needed for this one... We are given that $f$ is ...
1
vote
1answer
119 views

Matrix associated to a linear transformation

Today my linear algebra teacher explained what is the matrix associated to a linear transformation between two finitely generated $\mathbb{K}$-vector spaces. In particular, if we have $B = \{v_1, \...
0
votes
2answers
68 views

Show that $(z, w)$ is linearly dependent iff the imaginary part of $z\bar{w}$ is 0.

Consider $\mathbb{C}$ as $\mathbb{R}$-vector space. If $z,w \in \mathbb{C}$, show that $(z, w)$ is linearly dependent iff the imaginary part of $z\bar{w}$ is 0. I'm just unsure about the question and ...
5
votes
2answers
453 views

Basis for Tensor Product of Infinite Dimensional Vector Spaces

If V and W are vector spaces over a common field with bases $V_B = ${$v_i : i \in I$} and $W_B = ${$w_j : j \in J$}, then is {$v_i \otimes w_j: i \in I, j \in J$} a basis for $V \otimes W$ ? I have ...
2
votes
1answer
140 views

Find a basis for the subspace and state the dimension

${\{(a, b, c) : a-3b=0, b-2c=0, 2b-c=0}\}$ The answer states that there is no basis and that the dimension is 0, however I am unsure why. I suspect it is to do with all the equations equalling zero ...
0
votes
2answers
75 views

Find the bases relative in which TD matrix is in diagonal form

$D$ is the differentiation operator. $V$ is the linear space of all real polynomials of degree $\leq 3$. $T$ is the linear transformation that maps $p(x)$ to $xp'(x)$. $W$ is the image of $V$ under $...
0
votes
1answer
53 views

Norm of vector equals norm of it's basis representation

I will try to represent my question by example. There is a vector $a \in R^d$, basis $b$ spans $R^d$, so vector $a=\sum_{i=1}^{d}c_i b_i$. Whether $\left \| a \right \| = \left \| c \right \|$? If ...
0
votes
1answer
54 views

A question on vector spaces and their intrinsic properties

As far as I understand it, the notion of direction is not an intrinsic property of a vector in an abstract vector space; if the space is equipped with an inner product then one can determine angles ...
2
votes
3answers
140 views

Let $V,W$ be two countably infinite dimensional vector space over the same field , then are $V,W$ isomorphic as vector spaces?

Let $V,W$ be two countably infinite dimensional vector space over the same field , then are $V,W$ isomorphic as vector spaces ? And please give example of two non-isomorphic uncountable dimensional ...
0
votes
0answers
34 views

Basic Vector-space exercize

I've just begun studying groups, could you tell me if my answers to this exercice is correct ? We supply $ \mathbb{R}^{*+} $ with laws $ \forall \ x,y \ \in \ \mathbb{R}^{*+} \ x+y=xy \ $ (intern ...
0
votes
1answer
46 views

An irreducible action of an infinite cyclic group over a vector space

Let $\langle x\rangle$ be an infinite cyclic group and $V$ the additive group of an infinite vector space over $\mathbb{Z}_p$. Is it possible to make $\langle x\rangle$ act irreducibly on $V$? If ...
5
votes
1answer
232 views

What are the negative-dimentional n-sphere and n-cube?

The generalized formula for the volume and surface area of n-sphere allows to evaluate volumes and areas of negative-dimentional n-spheres. $$\begin{array}{ll} S_{n-1}(R) &= \displaystyle{\frac{n\...
0
votes
0answers
48 views

How to find generator matrix given a PCM of a Hamming code?

I'm having trouble as how to begin solving this. I'm given a Hamming Parity Check Matrix, and I have to find code vector V and generator matrix G. H = \begin{bmatrix} 0&0&1&1&0&1&...
0
votes
1answer
22 views

Canonical isomorphism $V^{*}/ Ann(W) \cong W^{*}$

I need to find a canonical isomorpism $V^{*} / Ann (W) \cong W^{*}$, $ W \subset V$ The first approach is: Let's consider a map $\phi: V^{*} \rightarrow W^{*}$, $ker( \phi)=Ann(W) $, then $V^{*} / ...
0
votes
1answer
59 views

Complexification and a canonical isomorphism

Let $W$ be a vector space over $\mathbb{R}$. How to build a canonical isomorphism $(W^{*})_{\mathbb{C}} \cong (W_{\mathbb{C}})^{*}$, where by $W_{\mathbb{C}}$ we denote a $W$ complexification. Would ...
1
vote
1answer
153 views

Test to know if a vector is inside a spherical triangle

Given a spherical triangle defined by $3$ unit vectors on a sphere, how can we test if a vector is contained inside the spherical triangle?
1
vote
1answer
49 views

What is the dimension of $\mathbb R[x] / \langle x^3-x\rangle$ as a vector space over $\mathbb R$ ?

What is the dimension of $\mathbb R[x] / \langle x^3-x\rangle$ as a vector space over $\mathbb R$ ? Can someone please give some links , articles where I can study about polynomila rings and its ...
1
vote
1answer
76 views

Question about bilinear pairing.

Let $V$ and $W$ be two $k$-vector spaces of dimension $n$ and let $\circ :V \times W \to k$ be a $k$-bilinear pairing that is nonsingular. If $\{v_1,..,v_n \}$ is a basis for $V$, how can I see that ...
-3
votes
2answers
64 views

Is $T(x,y)=(xy,0)$ a linear map? [closed]

Need to show that: $T = {R}^2 \rightarrow {R}^2 $ is not a linear transformation $T([ x, y]^ {T}) = [xy, 0]^ {T} = T $ Can you help get started.
0
votes
1answer
129 views

Hessian of Frobenius norm

I want to find the Hessian of the following function, $F(\mathbf{X}) = \frac{1}{2}||\mathbf{Y} - \mathbf{AX}||_F^2$. My try: Using trace formula for Frobenius norm, $F(\mathbf{X})$ can be written as, ...
1
vote
0answers
93 views

A Formal proof of Green Theorem

I want to go through the formal proof of Green theorem on a regular, simple and closed curve oriented counterclockwise and the vector space $F$ is a continuously differentiable vector field such that ...
-1
votes
3answers
65 views

If $\dim(V) = n$, is every spaning set $\{v_1,v_2,\ldots,v_n\}$ a basis for $V$?

Okay, so I need help clearing things up. Let $V$ be a vector space and $dim(V)=n$. Does it mean that every Spanning set $\{ v_1,v_2,v_3,\ldots,v_n \} $ is necessarily a basis for V? ...
2
votes
5answers
153 views

How to prove $ A^{\perp} $ is a closed linear subspace?

Suppose $ X $ is an inner product space and $ A\subseteq X $. I need to prove that $ A^{\perp} $ is a closed linear subspace of $ X $. Can anyone give me a idea?
0
votes
1answer
165 views

How Can I find the coordinates of a point, if I know its projection vector?

in this equation for vector projections, using the dot product $$ {{proj_\vec{a}\vec{b}} } = { \frac {\vec a \cdot \vec b}{\lVert\vec a \rVert^2} } \vec a $$ if I know everything BUT vector $\vec b$...
0
votes
2answers
585 views

Partial Derivative of a outer product in Vector Calculus

I am trying to compute the partial derivative of certain vector products for calculating the stiffness matrix. So we already know that For any vector $\textbf{x}$, we have 1) The derivative of the ...
2
votes
3answers
245 views

Relationship between a vector space V and it's dual space V*

I am studying relativity, and as you know the theory extensively uses the notion of covariant and contravariant component of vectors. My question is the following. Let say $\vec{x}$ is a vector ...
4
votes
1answer
145 views

Is $\mathbb{Z}_2$ a vector space?

If $\mathbb{Z}_2$ is defined in the usual way, is it a vector space? It passes $\vec{0}\in \mathbb{Z}_2$ and all the addition properties, my problem is with the scalar multiplication properties. Is $...
0
votes
1answer
68 views

Prove that $\dim V^*=\dim V$.

Let $V$ be a vector space over $F$, and let $V^*$ be the vector space of all linear functions from $V$ to $F$. Show that $\dim V^*=\dim V$. I was going over this problem with a friend, and he shared ...
1
vote
1answer
86 views

Other types of vector multiplication.

As far as I've seen, there are only two types of vector multiplication defined, dot product/inner product which is defined in a vector space of any dimension, even infinite, and cross product, which ...
0
votes
2answers
71 views

Is a vector space with two identical vectors a vector space with one or two vectors?

I'm new this, and cannot find any answers by searching. If a vector space has 2 identical vectors, in particular the zero vector, is it a vector space with 2 vectors or since they are linearly ...
2
votes
0answers
100 views

Let $V$ be a vector space. Prove that the doubleton set $S = \{u, v\}$ is linearly dependent if and only if $u$ is a scalar multiple of $v$.

I prove by contradiction. We assume that $S$ is linearly dependent, but $\vec{u}$ is not a scalar multiple of $\vec{v}$ and show that this leads to something completely and utterly ridiculous! If $\...
0
votes
2answers
110 views

Remind me: How do we convert coordinates to space with different basis and origin?

I'm stuck with the very beginning of my homework. I might've even misunderstood the task. I'll try to translate the assignment in case I fail to explain what I need later: Task description: ...
2
votes
2answers
1k views

How to prove a subspace is non empty?

To prove that a subspace W is non empty we usually prove that the zero vector exists in the subspace. But then is it necessary to prove the existence of zero vector. Can't we prove the existence of ...
0
votes
2answers
413 views

Proving that two lines are not from the same plane?

Well, I'm looking for a clean but effective way to prove that two lines in the space are or are not from the same plane, knowing that these two lines are given by their parametric representations.
0
votes
2answers
30 views

Method of proof confusing between Vector space and Linear Transformation.

I'm confusing about the way to determine the Vector space and the Linear Transformation. My knowledge is the way to determine whether the map given is Linear Transformation by proving this : $T(\...
0
votes
1answer
39 views

Basis vectors of a linear space

Knowing that: $$W=\{(x_1,x_2,x_3,x_4) \in \mathbb{R}^4 | x_1+2x_2-x_3+x_4=0 \}$$ Is the basis of this subspace $(-2,1,0,0),(1,0,1,0),(-1,0,0,1)$ If it is true then i need only the simple yes/no answer....
2
votes
2answers
846 views

Formula to best fit a rectangle inside another by scaling

I am very week in Math. I am a web programmer, and usually my work does not involve too much math - its more of putting records into database, pulling out reports, making those fancy web pages etc etc....
1
vote
1answer
48 views

$L_1 \cap L_2$ is dense in $L_2$?

We were talking about Fourier series the other day and my professor said that the requirement that a function be in $L_1 \cap L_2$ wasn't a huge obstacle, because this is dense in $L_2$. Why is this ...