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
157 views

Find a matrix such that $Ax=0$

Let $$W = span\left\{ {\left( {\matrix{ 1 \cr 0 \cr 0 \cr 1 \cr } } \right),\left( {\matrix{ 0 \cr 2 \cr 1 \cr { - 1} \cr } } \right)} \right\}$$ I was asked ...
2
votes
1answer
261 views

Proving linear independence

Let $A$ be an $n \times n$ matrix and suppose $v_1, v_2, v_3 \in \mathbb{R}^n$ are nonzero vectors that satisfy: $$ Av_1 = v_1 \\ Av_2 = 2v_2 \\ Av_3 = 3v_3 $$ Prove that $\{v_1, v_2, v_3\}$ is ...
2
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1answer
1k views

Finding a basis for the intersection of two subspaces

I'll try to write this as best as I can... Let the following $U_1, U_2$ be subspaces of $\mathbb{R}^4$ $$ U_1 = \begin{Bmatrix} (x, y, z, w) : z-y+2w = 0 \end{Bmatrix} $$ $$ U_2 = ...
1
vote
1answer
106 views

Is $\mathbb{R^Z}$ or its elements countable?

Continue on the self study on infinite vector spaces. According to this link, $\mathbb{R^Z}$ has elements of the following form: $$(y_k)_{k\mathbb{\in Z}}=(\dots y_{-1},y_0,y_{1}\dots)$$ Or more ...
1
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1answer
37 views

Find a isometry such that the matrix in respect to the canonical basis is:

I need to find a isometry such that the matrix in respect to the canonical basis is: $$\begin{bmatrix}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0\\0 & 0 & 1\\x & y & ...
1
vote
3answers
57 views

Finding a basis for a certain vector space of periodic polynomials

I am having a little bit of trouble solving an homework question. I found that $S={ p(x) \in R_4[x]} \big| p(x)=p(x-1) $ is a vector space. Now I need to find some set, K that holds ${span(k)=S}$ ...
1
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2answers
60 views

Proof that $(a+b, a)$ belongs to $\mathbb{R}^2$

I am aware that the vector $(a+b, a)$ such that $a$, $b$ are real numbers belongs to $\mathbb{R}^2$, which is defined by any vectors $(x_1, x_2)$ such that $x_1, x_2$ are real numbers. Is there a way ...
1
vote
0answers
170 views

Another proof of uniqueness of identity element of addition of vector space

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is solely based on vector space axioms. Axiom ...
1
vote
2answers
107 views

Linear Space in Vector Spaces question.

How do I do the first part of the question where they say V1 U V2 is not a linear space, please help my exam is very close. In the marking scheme it says it's not closed under addition but can ...
1
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2answers
4k views

Do four dimensional vectors have a cross product property? [duplicate]

We know how to make cross product of three dimensional vectors. $$ \vec A \times \vec B = \vec C$$ Where : $ \vec A = (A_i; A_j; A_k)$ $ \vec B = (B_i; B_j; B_k)$ $ \vec C = (C_i; C_j; C_k)$ $C_i = ...
1
vote
2answers
73 views

Unitary map between sets of vectors

Suppose I have two sets of vectors, $E_1=\{v_i\}_{i=1}^{k}$ and $E_2=\{u_i\}_{i=1}^{k}$, with each vector belonging to $\mathbb{C}^k$. When is it possible to find a unitary matrix that maps $E_1$ to ...
1
vote
2answers
95 views

$T\circ T=0:V\rightarrow V \implies R(T) \subset N(T)$

Question Let $T:V \rightarrow V$ be a linear map. How do I prove that $T \circ T = T_0$ ( the zero linear map) iff $R(T) \subset N(T)$? Attempt \begin{eqnarray} T\circ ...
1
vote
1answer
79 views

Dimension Recovery of $S \subset P_n(F)$

How is the subset of $P_n(F)$ consisting of all polynomials $f$ such that $f(1) = 0$ a subspace of $P_n(F)$? What is the dimension of this subset? Added from answer posted by Trancot on 18 Apr ...
1
vote
2answers
137 views

Calculating new vector positions

I'm using the following formula to calculate the new vector positions for each point selected, I loop through each point selected and get the $(X_i,Y_i,Z_i)$ values, I also get the center values of ...
1
vote
3answers
6k views

Finding the Angle theta between two 2d vectors.

Find the angle α between the vectors $$\begin{bmatrix}3 \\ 5 \end{bmatrix}and \begin{bmatrix}-2 \\ 3 \end{bmatrix} $$ I found 64.654, but apparently is wrong from what the webwork says. Can anyone ...
1
vote
2answers
3k views

Calculate intersection of vector subspace by using gauss-algorithm

There are two vector subspaces in $R^4$. $U1 := [(3, 2, 2, 1), (3, 3, 2, 1), (2, 1, 2 ,1)]$, $U2 := [(1, 0, 4, 0), (2, 3, 2, 3), (1, 2, 0, 2)]$ My idea was to calculate the Intersection of those two ...
1
vote
1answer
143 views

Prove: If $L \leq X$, $L$ has finite dimension, $M\leq X$ Then $L+M$ is closed.

Prove: If $X$ is a locally convex space, $L \leq X$, $L$ has finite dimension, $M\leq X$ Then $L+M$ is closed. What I know: If $L$ is a finite dimensional subspace, then $L$ is closed.
1
vote
1answer
63 views

Linear basis of sum of kernels of two linear applications from $\mathbb R^4$ to $\mathbb R^2$

Let $$L_{1}(x_{1},x_{2},x_{3},x_{4})=(3x_{1}+x_{2}+2x_{3}-x_{4}, 2x_{1}+4x_{2}+5x_{3}-x_{4})$$ and $$L_{2}(x_{1},x_{2},x_{3},x_{4})=(5x_{1}+7x_{2}+11x_{3}+3_{4}, 2x_{1}+6x_{2}+9x_{3}+4x_{4})$$ Let ...
1
vote
1answer
732 views

Could intersection of a subspace with its complement be non empty.

If that is possible could you please correct my understanding about complement of a subspace. From what i recall from set theory. A complement of a set B is the set U - B where U is the universal ...
1
vote
2answers
388 views

First Order Language for vector spaces over fields

I'm attempting to come up with a first order language $L$ that is able to describe vector spaces over fields. I came up with a few sets of nonlogical symbols. $Rs_L=\{Scal,Vec\}$ where $Scal,Vec$ are ...
0
votes
2answers
50 views

Suppose $S_1 =\{ u_1 , u_2 \}$ and $S_2 = \{ v_1 , v_2 \}$ are each independent sets of vectors in an n-dimensional vector space V.

Let us assume that every vector in $S_2$ is a linear combination of vectors in $S_1$. Question: Does that mean that $S_1$ and $S_2$ are bases for the same subspace of $V$? I know that the answer to ...
0
votes
1answer
47 views

Spans of Orthogonal complements

Let $A$ be the matrix $$ \begin{pmatrix} 1 & 1 & -1&-1 \\ 1 & 2 & -2 & 1 \\ \end{pmatrix} ,$$ let $W$ = ker $A$ and let $W^\bot$ be the ...
0
votes
1answer
62 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 $\{i,j,k\}$ ...
0
votes
1answer
36 views

Vector spaces and infinite cyclic linear transformations II

Let $V$ be the vector space of infinite dimension on the field $\mathbb{Z}_2$. Let's say $$ V=\langle v_1\rangle+\langle v_2\rangle+\dots+ \langle v_n\rangle+\dots$$ where each $v_i$ has order $2$ and ...
0
votes
1answer
62 views

Given a field extension $K\colon F$, $K$ is an $F$-vector space

I'm having a hard time understanding fields. Could someone help with the following I need to show that if $F$ $\subseteq$ $K$ are both fields and addition and multiplication on F are the ...
0
votes
1answer
109 views

Determining a basis of vector space with unusual addition and multiplications: $(a,b)+(c,d)=(ad+bc,bd) $ and $k * (a,b) =(kab^{k-1},b^k)$

Let $\mathbb W=\{(a,b) \in \mathbb R^2 \mid b>0\}$ and define addition by $(a,b)+(c,d)=(ad+bc,bd) $ and define scalar multiplication by $k * (a,b) =(kab^{k-1},b^k)$. Find a basis for $\mathbb ...
0
votes
1answer
71 views

Having trouble with a linear lager question about kernel and basis

Let $V$ be a $2$-dimensional vector space, and let $\alpha={e_1,e_2} $ be a basis for $V$. Define a linear transformation $T: V\to V$ by declaring that: $T(e_1+e_2)=2e_1−e_2 $ $T(e_2)=4e_1−2e_2$. ...
0
votes
2answers
4k views

matrices forms a basis for vector space 2x2

$\begin{bmatrix}0&1\\2&3\end{bmatrix}$ $\begin{bmatrix}3&4\\5&6\end{bmatrix}$ $\begin{bmatrix}7&8\\9&10\end{bmatrix}$ $\begin{bmatrix}11&12\\13&14\end{bmatrix}$ Show ...
0
votes
2answers
349 views

A subspace contains the zero vector; intersection of subspaces is a subspace

I have a simple question I have to answer but I am not sure where to start with this due to my lack of experience regarding subspaces. Can anybody help me? Assume $V \subset \Bbb R^n$ is a ...
0
votes
1answer
106 views

Find where $r(t)=<t,t,t^2>$ hits the $x-y$ plane

I have to find $r'(t)$ and $||r'(t)||$ for $r(t)=<t,t,t^2>$, which I know how to do. $r'(t)=<1,1,2t>$ $||r'(t)||=\sqrt{2+4t^2}$ The problem is that my professor didn't explain how to ...
0
votes
2answers
2k views

Proof help: The span of any list of vectors in V is a subspace of V.

I'm having some trouble understanding this proof. Let $U = ()$, the empty set, and define span$(U) = 0 \subset V$. Now let $U = (v_1 \cdot \cdot \cdot v_n)$ be a list of vectors in $V$. This means ...
0
votes
1answer
269 views

How do you prove $\def\rank{\operatorname{rank}}\rank(f_3 \circ f_2) + \rank(f_2 \circ f_1) \leq \rank(f_3 \circ f_2 \circ f_1) + \rank(f_2) $? [duplicate]

Possible Duplicate: Lower bound involving the rank of the composition of linear transformations The following question is about a lower bound on the rank of a composition of functions given ...
0
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2answers
338 views

Lower bound involving the rank of the composition of linear transformations

The following question is about a lower bound on the rank of a composition of functions given as a simple expression for the two terms of the sum involved in the inequality. Consider ...
29
votes
4answers
1k views

Given two basis sets for a finite Hilbert space, does an unbiased vector exist?

Let $\{A_n\}$ and $\{B_n\}$ be two bases for an $N$-dimensional Hilbert space. Does there exist a unit vector $V$ such that: $$(V\cdot A_j)\;(A_j\cdot V) = (V\cdot B_j)\;(B_j\cdot V) = 1/N\;\;\; \ ...
15
votes
3answers
56k views

How to tell if a set of vectors spans a space?

I want to know if the set $\{(1, 1, 1), (3, 2, 1), (1, 1, 0), (1, 0, 0)\}$ spans $\mathbb{R}^3$. I know that if it spans $\mathbb{R}^3$, then for any $x, y, z, \in \mathbb{R}$, there exist $c_1, c_2, ...
16
votes
1answer
760 views

Why is the inclusion of the tensor product of the duals into the dual of the tensor product not an isomorphism?

Let $V$ and $W$ be vector spaces (say over the reals). There is a linear injection $V^* \otimes W^* \to (V \otimes W)^*$ which sends $\sum_i f_i \otimes g_i \in V^* \otimes W^*$ to the unique ...
17
votes
3answers
1k views

What is an inner product space?

As I've understood it, what I've learned is that the dot product is just one of many possible "inner product spaces". Can someone explain this concept? When is it useful to define it as something ...
14
votes
1answer
760 views

Why it is important for isomorphism between vector space and its double dual space to be natural?

I'm reading the book (by A. Kostrikin) on linear algebra and I feel like I'm really missing something about this idea. I understand the formal proofs of: a) isomorphism between vector space $V$ and ...
9
votes
1answer
1k views

Vector, Hilbert, Banach, Sobolev spaces

Trying to wrap my head around all these different spaces. Which one is the most general? Can you summarize the differences between them? Is there a notable space that I missed?
10
votes
2answers
2k views

Can a basis for a vector space be made up of matrices instead of vectors?

I'm sorry if this is a silly question. I'm new to the notion of bases and all the examples I've dealt with before have involved sets of vectors containing real numbers. This has led me to assume that ...
7
votes
4answers
3k views

*understanding* covariance vs. contravariance & raising / lowering

There are lots of articles, all over the place about the distinction between covariant vectors and contravariant vectors - after struggling through many of them, I think I'm starting to get the idea. ...
15
votes
1answer
1k views

Prove $\mathbb{Z}$ is not a vector space over a field

This is an exercise from Chapter 3 of Golan's linear algebra book. Problem: Show $\mathbb{Z}$ is not a vector space over a field. Solution attempt: Suppose there is a such a field and proceed by ...
14
votes
2answers
519 views

Geometric interpretation of the cofactor expansion theorem

I find the geometric interpretation of determinants to be really intuitive - they are the "area" created by the column vectors of the matrix. Could someone give me a geometric interpretation of the ...
11
votes
2answers
1k views

Do I understand metric tensor correctly?

So I've been studying vectors and tensors, and I'm trying to understand metric tensors. As I understand them, besides a vast array of explanations, they provide an invariant distance between vectors ...
9
votes
2answers
12k views

What is the proof that covariance matrices are always semi-definite?

Suppose that we have two different discreet signal vectors of $N^\text{th}$ dimension, namely $\mathbf{x}[i]$ and $\mathbf{y}[i]$, each one having a total of $M$ set of samples/vectors. ...
8
votes
1answer
1k views

Do all vectors have direction and magnitude?

I go by Vector. It's a mathematical term, represented by an arrow with both direction and magnitude. Vector! That's me, because I commit crimes with both direction and magnitude. Oh yeah! For ...
7
votes
2answers
328 views

Finite-dimensional space naturally isomorphic to its double dual?

The example that a finite vector space is naturally isomorphic to its double dual seems to be the canonical example of natural isomorphisms. Concretely, there are two functors $\mathsf{Id}, {-^*}^* : ...
6
votes
2answers
3k views

Relation between cross-product and outer product

If inner products ($V$) are generalisations of dot products ($ \mathbb{R}^n$), then are outer products ($V$) also related to cross-products ($ \mathbb{R}^3$) in some way? A quick search reveals that ...
5
votes
2answers
210 views

Cross products?

Say you have vectors $v$ and $w$. Let there cross product be denoted by $x$ so that: $$v \times w = x$$ According to Wikipedia: $$x_x = v_yw_z - v_zw_y$$ $$x_y = v_zw_x - v_xw_z$$ $$x_z = v_xw_y - ...
5
votes
0answers
136 views

Why is the radical of a Clifford algebra generated by the kernel of the associated symmetric form? [duplicate]

I was recently reading through Jacobson's Basic Algebra. I got to the section on Clifford algebras, and have the following question. Let $Cl_\omega$ be the Clifford algebra with bilinear symmetric ...