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 ...

learn more… | top users | synonyms

0
votes
0answers
38 views

Is there a linear transformation that sends v to (1,0)

Suppose we are given a linear vector field $v(x) = Ax$ on $\Bbb R^2$. Suppose $x_0$ is the point where this vector field does not vanish. Is it possible to find a linear transformation from $\Bbb R^2$ ...
1
vote
0answers
17 views

Set of planes to create a geometric plane

How do I come up with a set of planes that will create a geometric shape? The shape can contain up to 40 sides. Is there any general formula/equation to solve this kind of question?
0
votes
2answers
27 views

Find two vectors that make an angle of $60^\text{o}$ with $\vec v= \langle 3,4 \rangle$

[I'm possibly going against the site's etiquette by doing this, but I'm aware that this question has already been asked- the reason I'm posting this new one is that (1) it seems that the original ...
0
votes
0answers
17 views

adding constant functions to an algebra

Let $C_b(D)$ be the space of bounded continous functions from the metric space $D$ to $\mathbb{R}$. Let $F \subset C_b(D)$ be an algebra that separates points of $D$. In a proof I am reading at the ...
0
votes
1answer
49 views

What does the notation $\mathbb R[x]$ mean?

What does the notation $\mathbb R[x]$ mean? I thought it was just the set $\mathbb R^n$ but then I read somewhere that my lecturer wrote $\mathbb R[x] = ${$\alpha_0 + \alpha_1x + \alpha_2x^2 + ... + ...
1
vote
1answer
75 views

Dimension of union of fields over intersection of fields.

What is the dimension of the union of fields $L_1$ and $L_2$ over $L_1 \cap L_2$ ($L_1$ and $L_2$ have dimensions $n_1$ and $n_2$ over $L_1 \cap L_2$)as a vector space? I think it should be $n_1n_2$ ...
0
votes
1answer
19 views

Equality with rank and dimension of linear map (endomorphirms).

I am desperately looking for a hint ! I have to prove the following equality : $\forall (u, v) \in \mathscr{L}(E)^2, \mathrm{rg} (v) - \mathrm{rg}(u \circ v) = \dim (\mathrm{Ker} (u) \cap ...
0
votes
0answers
23 views

Is it true that every two Hamel Basis for a Vector Space have the same cardinality? [duplicate]

Let $X$ be a vector space over any field $F$, and let $\mathcal{B}$ and $\mathcal{C}$ Hamel basis for $X$. My question is Is there a bijection $\phi : \mathcal{B}\to \mathcal{C}\ ?$ i.e. do ...
1
vote
1answer
41 views

Vector Space: is there a general definition of annihilator beyond the dual space?

The definitions I see for the annihilator of a subset S of a vector space V over a field F is the subset $S^0$ of the dual space $V^*$given by $S^0 = \{\varphi\in V^\star:\varphi(S)=0\}$, and the ...
1
vote
1answer
20 views

Kernel and Image (of endomorphism) inclusions

Are these propositions always true (for any $u \in \mathscr{L}(E) \neq 0$) ? $\mathrm{Ker}(u) \subsetneq \mathrm{Ker}(u^2)$ $\mathrm{Im}(u^2) \subsetneq \mathrm{Im}(u)$ If it is, how can I prove ...
0
votes
1answer
25 views

Check whether a system {$v_1,…,v_m$} of vectors in $\mathbb R^n$ (in $\mathbb R[x]$) is linearly independent.

Check whether a system {$v_1,...,v_m$} of vectors in $\mathbb R^n$ (in $\mathbb R[x]$) is linearly independent. These are my thoughts: For {$v_1,...,v_m$} to be linearly independent, prove that: ...
0
votes
0answers
10 views

Proving a relation in a vector space

I have the following velocity vector (in spherical polars): \begin{equation} \textbf{v} = u \hat{\textbf{r}} + v_{\phi}\hat{\boldsymbol\phi} \end{equation} Where $u(r) = u$ and $v_{\phi} (r) = ...
1
vote
2answers
37 views

What is the rigorous justification for using inner products as a function of similarity between two vectors?

In machine learning, it is a common thing to define similarity measures, specially using the so call Kernel function. Kernel functions are defined though through inner products of feature vectors: ...
0
votes
0answers
19 views

All rank two symmetric tensors are several conformally flat metrics summed together?

If given a rank two symmetric tensor $T_{mn}$ can I decompose it as \begin{align} T_{mn} = \sum_{i=1}^{M} \phi_ig_{mn}{}^{i}{} \end{align} where $\phi_i$ are the $i$th conformal factors and ...
1
vote
1answer
31 views

3d vector perpendicular confusion

This is a very specific issue I'm dealing with, so I'll post the question below: The Diagram [not included] illustrates the flight path of a helicopter H taking off from an airport. Coordinate ...
1
vote
1answer
37 views

Checking if the set of solutions to $y''+ 4y'+ 8y = 0$ is a vector space with the usual operations.

I've read a question that ask me to check that the set of functions on the line that has second derivative and verify the equation $y''+ 4y'+ 8y = 0$ is a vector space with the usual operations. I ...
-1
votes
1answer
24 views

Showing that following is not a vector space?

I have the following 8 axioms for a Vector Space and the following question. I managed to prove that Axiom 3 doesn't work(and as a result Axiom 4 because 0 element doesn't exist) but the answer ...
4
votes
0answers
130 views

The dimension of vector space $F^{X}$.

Here's the problem: Let $F$ be field, $X$ an infinite set and $F^{X}$ be the set of all functions $f:X\rightarrow F$. Then $F^{X}$ is a vector space over $F$ (with $(f+g)(x)=f(x)+g(x)$ and ...
0
votes
2answers
114 views

Integration over manifolds

S is subset of $\Bbb R^3 $ consisting of the union of 1) $z$ axis 2) the unit circle $x^2+y^2=1,z=0$ 3) the points $(0,y,0)$ with $y \ge1 $ Let $A$ be the open set $\Bbb R^3-S$. Let $C_1, C_2, ...
0
votes
2answers
32 views

What does component being zero in particular dimension mean?

I have asked a question on stack physics , basically asking why is it so that every $4\times 1$ matrix can't be written as tensor product of two $2\times1$ matrices ? (for more detail : 4-D column as ...
0
votes
0answers
8 views

Direction Ratio Mapping

I reading a code where Normal vector to a plane is given. then a,b,c are taken (what I guess is direction ratio values). a=norm_vec(1); b=norm_vec(2); c=norm_vec(3); Now I cannot understand How ...
1
vote
2answers
28 views

Dimension of a subspace smaller than dimension of intersection

Suppose I have a finite-dimensional vector space $V$, and $U_1, U_2$ are subspaces of $V$, such that $U_1\nsubseteq U_2$. Is it possible that $\dim{U_1}\leq\dim{U_1 \cap U_2}$?
1
vote
0answers
21 views

How to convert from cartesian to vector form (of a straight line)

line in question: $$-x - 1 = \frac 12y - \frac12 = \frac 12z + 1$$ I wasn't really sure how to go about this one since it's not in the exact general form that I was taught, but I went along and ...
-1
votes
1answer
42 views

What are the objects and morphisms of the category $\operatorname{Vect}$?

What are the objects and morphisms of the category $\operatorname{Vect}$? I am trying to learn category theory, and it seems we have infinite objects in $\operatorname{Vect}$ being all of the finite ...
0
votes
1answer
19 views

Vector inequality for a scalar difference of two vectors in $\mathbb{R}^n$.

A student posed an interesting problem to me the other day and embarrassingly I could not prove or disprove it even though it appears relatively simple. The question was: Given vectors ...
0
votes
0answers
22 views

Vector space multiplication matrix

I have to find normal basis for field $GF(3^6)$ which represents vector space over field $GF(3^2)$ and then find multiplication matrix for this vector space. Then I have to demonstrate fast ...
1
vote
1answer
31 views

Finding normal basis for GF(q^m) over GF(q)

Could you kindly explain, how can one find a normal basis for GF$(3^6)$ over the GF$(3^2)$? As I understood, I should start with finding the polynomial in a form $$a(x^2) + (a^9)x + a^{81},$$ which ...
1
vote
3answers
296 views

Does $\{y\in \mathbb{R}^n:\operatorname{rank}((x,y,Ay))=2\}$ have zero Lebesgue measure?

This is probably a simple question, but I need some help. Consider a vector $x\in \mathbb{R}^n$ and a real $n\times n$ matrix $A$. I'm interested in the set of $y\in\mathbb{R}^n$ such that $x,y,Ay$ ...
0
votes
2answers
28 views

Basis 'vectors', basis 'matrices'?

Let $\mathfrak{sl}_2$ be the vector space of $2\times 2$ traceless matrices. Let $A\in \mathfrak{sl}_2$ be a diagonal matrix. Define a linear operator: $$\phi_A(X)=AX-XA$$ $\phi_A:sl_2\to sl_2$ What ...
0
votes
1answer
37 views

Null spaces associated to eigen values

Suppose that $f$ is an endomorphism in a finite dimensional vector space, and $\lambda$ is an eigenvalue of $f$. Let $C_f = \text{det}(f - X \,\text{Id})$ be the characteristic polynomial of $f$, and ...
1
vote
2answers
49 views

How to represent matrix as the sum of rank-one matrices

If we're given $B$ to be a $4 \times 7$ matrix: $$\begin{bmatrix}1 & 2 & -3 & 7 & 0 & -2 & 5\\1 & 2 & -3 & 7 & 1 & 3 & -2\\0 & 0 & 0 & 0 ...
1
vote
1answer
37 views

Dimension of a subspace of $\operatorname{Hom}(\mathbb{R}^3, \mathbb{R}^4)$

Let $V = \{f \in \operatorname{Hom}(\mathbb{R}^3,\mathbb{R}^4) \mid f(x,y,z)=f(z,x,y)\}.$ Find $\dim_\mathbb{R}(V)$. I think I need to use the dimension theorem for vector spaces: I need to ...
1
vote
0answers
44 views

How to calculate the dimension of an infinite direct product of copies of a field?

Let $F$ be a field and $I$ an arbitrary infinite index set. I'd like to know how to calculate the dimension of $\prod_{i\in I}F$. By the way, I know $\dim(\prod_{i\in I}F)\geqslant ...
14
votes
4answers
1k views

How to develop an intuitive feel for spaces

I'm a physicist who's currently delving deeper into what I would call more 'hardcore' maths (e.g. FEM and control theory). Every now and then, I come across various spaces, such as vector spaces, ...
1
vote
1answer
27 views

If $T:\mathbb{R}_3→\mathbb{R}_7$ is a linear transformation, then is the set $\operatorname{ker}(T)$ a subset of the codomain?

If $T:\mathbb{R}_3→\mathbb{R}_7$ is a linear transformation, then is the set $\operatorname{ker}(T)$ a subset of the codomain? My answer is 'yes', because $\operatorname{ker}(T) \subset ...
0
votes
2answers
35 views

Basis for the null space of an identity matrix

Is the set containing only the zero vector a basis for the null space of an identity matrix?
0
votes
1answer
26 views

Linear dependence when number of vectors is greater than/less than the dimensions of the vector space

Simple question here, I just need some clarification of a theorem. Theorem: if k > n, then any k vectors in $R^n$ are linearly dependent. Nice and easy I guess! My question is this: Does this imply ...
2
votes
1answer
44 views

What is the probability of choosing r independent vectors in $\mathbb{R}^n$ in the unit sphere?

I was trying to compute the probability of choosing $r \leq n$ indepedent vectors $a_i \in \mathbb{R}^n$ such that they are independent. I was told that the probability that they are not independent ...
0
votes
1answer
24 views

Is this a vector space? If not, how can I make it one?

S (2x2 matrix) = {(a, b), (c,1) | $a,b,c$ is in $ \Bbb R$} I know for a vector space we must: 1. Define Addition 2. Define Scalar Multiplication 3. Have a set of numbers 4. Have a Field I know that ...
2
votes
3answers
59 views

What is abstraction of direction in considering vectors such as used in Engineering & Physics?

In the use of vectors of engineering and physics, we encounter objects that obey the axioms of a vector space but also have two new attributes of length (or, magnitude) and direction (e.g. direction ...
0
votes
1answer
17 views

Find a finite set of vectors which spans $W$.

Let $W$ be the set of all $(x_1, x_2, x_3, x_4, x_5)$ in $\Bbb R^5$ which satisfy $2x_1-x_2+{4 \over 3}x_3 - x_4\qquad = 0$, $x_1\qquad+{2 \over 3}x_3\qquad- x_5 = 0$, $9x_1-3x_2+6x_3-3x_4-3x_5 = 0$. ...
2
votes
1answer
22 views

Prove $V_{e} + V_{o} = V$

Prove $V_{e} + V_{o} = V$ where $V_{e}$ is a subset of even functions from $R$ into $R$, $V_{o}$ is a subset of odd functions from $R$ into $R$. I have proved $V_{e}$, $V_{o}$ are subspaces and ...
1
vote
1answer
17 views

Show that the vectors form a basis for $R^3$.

Show that the vectors $\alpha_1 = (1, 0, 1)$, $\alpha_2 = (1, 2, 1)$, $\alpha_3 = (0, -3, 2)$ form a basis for $R^3$. Is it enough to show that the vectors are linearly independent?
0
votes
3answers
25 views

Gram-Schmidt Process and Orthogonal Components

Let the Gram-Schmidt process transform the vector system $(a_{1}, ..., a_{n})$ into the system $(b_{1}, ..., b_{n})$. Show that the vector $b_{k}$ is the orthogonal component of the vector $a_{k}$ ...
0
votes
0answers
21 views

Proving properties about cofactors when matrix is not invertible

(1) $cof(A^t) = cof(A)^t$ (2) $cof(A)^t = det(A)I$ I have (at least, I think so) proofs of (1) and (2). But the proofs require the matrix $A$ to be non-singular. How do I prove (1) and (2) if $A$ is ...
-1
votes
1answer
29 views

An example of centrally symmetric unbounded set in $\mathbb{R}^2$ which is convex? [closed]

Can you find an example of a centrally symmetric, unbounded, convex set in $\mathbb{R}^2$ which does not contain points of the form $\left\{(m,n)|\ m,n \in \mathbb{Z}\ \ (m,n) \neq (0,0) \right\}$ ?
0
votes
1answer
46 views

Gram-Schmidt procedure on functions

I have been applying the Gram-Schmidt procedure with great success however i am having difficulty in the next step, applying it to polynomials. Here i what i understand If i have 2 functions, say ...
0
votes
0answers
19 views

Problems with the hypothesis in a fixed point theorem

Leray-Schauder fixed point theorem : If $D$ is a non-empty , convex , bounded and closed subset of Banach space $B$ and $T:D \to D$ a compact map , then $T$ has a fixed point in $D$. I have ...
2
votes
0answers
62 views

What is the $\dim L(X,Y)$?

Let $X$ and $Y$ be two finite-dimensional vector spaces over the same field $K$, and let $L(X,Y)$ denote the vector space of all linear operators $T \colon X \to Y$. Then what is $\dim L(X,Y)$? My ...
1
vote
3answers
71 views

If M,N are finite dimensional vector spaces with same dimension ,then if M is subset of N ,then M=N

If M,N are finite dimensiona;l vector spaces with same dimension then if M is subset of N ,then M=N I think i need to show that both vector spaces are spanned by same bases in order to do this or to ...