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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If commutativity of vector space is omitted, can we still use other axioms to prove the commutativity?

Here I am thinking of using $-(x+y)$ and show that it equals $-(y+x)$. $-(x+y)=-x-y$ by distributivity =$-x-y+0=...$ Here I don't know how to continue, could someone suggest?
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1answer
14 views

How to show that the following is satisfied for all vector space axiom?

Let $V=\{a_2x^2+a_1x+a_0|a_1, a_2, a_3\in \mathbb{R}, a_2\ne 0\}$ with operation defined by $$(a_2x^2+a_1x+a_0)+(b_2x^2+b_1x+b_0)=(a_2+b_2)x^2+(a_1+b_1)x+(a_0+b_0)$$ ...
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1answer
21 views

How to show that $M_{2\times 2}(\mathbb{R})=W_1\oplus W_2$ based on the following assumption?

Let the subspaces $W_1=\{\begin{pmatrix}a&b\\-b&a \end{pmatrix}|a, b\in \mathbb{R}\}$ and $W_2=\{\begin{pmatrix}c&d\\d&-c \end{pmatrix}|c, d\in \mathbb{R}\}$ of $M_{2\times ...
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1answer
9 views

If $x+y=(x_1y_1, …, x_ny_n)$ and $c\cdot '\ x=x^c_1, …, x^c_n$, how to show that with these two operation $V$ is a subspace?

Let $V=(R^+)^n=\{(x_1, ..., x_n)| x_i\in R^+$for each $i\}$. In $V$ define a vector sum operation $+'$ by $x+y=(x_1y_1, ..., x_ny_n)$ and scalar multiplication $\cdot '$ by $c\cdot '\ x=x^c_1, ..., ...
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2answers
19 views

How to show if the following subset $W$ is a subspace of a vector space $V$?

$1.$ $V=P_n(\mathbb{R}), $and $ W=\{p(x)\in P_n(\mathbb{R})\mid p(1)+p(2)+p(3)=0 \}$ $2.$ $V=M_{n\times n}(\mathbb{R}), $and $ W=\{A\in M_{n\times n}(\mathbb{R}) \mid A \text{ is not symmetric}\}$ ...
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1answer
16 views

Given a basis $U$, what conditions are needed for an orthogonal basis for it?

Given a basis $U$, what conditions are needed for an orthogonal basis for it? For example, in the following vector space $U$, if $U =sp\{(1,1,1),(1,3,7)\}$ then what conditions are needed for an ...
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0answers
7 views

how to write amatlab code for document representation using second order tensor [on hold]

My project is on document representation using tensor.How i will represent a document using second order tensor in matlab?
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2answers
36 views

Linear transformation from a vector space to a field

Can anyone help me with the following question: Let $V$ be a vector space over field $F$, possibly not finite dimensional. Let $T \colon V \to F$ be a linear map. Prove that there is no subspace ...
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1answer
26 views

Showing that the following vectors are linearly independent in a subspace which they do not span.

I am trying to better understand vector spaces and dimensions. I could prove (i) via induction and the definition of linear independence? However how can I approach the questions (ii),(iii) which ...
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3answers
36 views

The union of three subspaces equals to a vector space

I am aware that the union of subspaces does not necessarily yield a subspace. However, I am confused about the following question: (i) Let $U, U'$ be subspaces of a vector space $V$ (both not ...
2
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3answers
25 views

Proof: dimension of the vector space of solutions to the system Bx=0

I've run into a matrix dimension proof I'm having some trouble with: Let $A,B$ be $n\times n$ matrices, and let $P(A),P(B),P(AB)$ be the vector spaces of solutions to the systems ...
3
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2answers
32 views

Finding ranks and nullities of linear maps

I am confused about ranks, nullities and bases of the kernel. From what I understand the rank is the dimension of a vector space generated by a matrix. How would I do the following examples? Find ...
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2answers
22 views

Describe all vectors $v = \pmatrix{x\\y}$ that are orthogonal to $u = \pmatrix{a\\b}

Describe all vectors $v = \pmatrix{x\\y}$ that are orthogonal to $u = \pmatrix{a\\b}$. I know that vectors that are orthogonal will have a dot product of 0. So here's what I was thinking: ...
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2answers
25 views

Finding a linear transformation with a given null space

The problem statement is, Find a linear transformation $T: \mathbb R^3 \to \mathbb R^3$ such that the set of all vectors satisfying $4x_1-3x_2+x_3=0$ is the (i) null space of $T$ (ii) range of $T$. ...
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1answer
23 views

Relating regression to projection?

I recently learned that one can think of regression as a projection of a vector in a high dimension space onto the other vector. I tried implementing this and got it to work: ...
2
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1answer
29 views

When is a manifold also a vector space?

My question arises from this definition: Poincare group is the group of Minkowski space-time isometries. Which means that it leaves the space-time intervals unchanged. Now here is my understanding: ...
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1answer
25 views

Dimension of the subspace of a vector space spanned by the following vectors.

I know that in order to find a subsequence that is a basis of a subspace is to check whether the given vectors are linearly independent and whether they span the subspace. However how can I find the ...
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1answer
24 views

Independence, Inverse and Additive Identity for vector with defined vector addition and scalar multiplication

I am struggling to find my bearings on this question. I am confident that I can do parts a and d. I have no clue how to approach b. I was also wondering if the redefined vector addition and scalar ...
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1answer
13 views

Write $F$ as a linear combination of elements of $\mathcal B^*$

If $V=\mathbb R[x]_k=\{\sum\limits_{i=1}^ka_ix^i:a_i\in\mathbb R, \forall i\}$ is a vector space of dimension $k+1$ over $K=\mathbb R$ and $\mathcal B=\{1,x,\dots,x^k\}$ is a basis of $V$. The dual ...
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2answers
38 views

Describing all the linear transformations satisfying the constraints

How to find the linear transformation $T: \mathbb R^3 \to \mathbb R^3$ such that the set of all vectors satistfying $4x_1-3x_2+x_3=0$ is a) Null space of $T$ b) Range of $T$ I'm not able to ...
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1answer
23 views

Proving a basis spans $R^3$

Doing some reviewing and I'm not 100% sure if my thought-process is correct. I have the following two vectors and need to prove they're a basis for $R^3$: $$B= \begin{bmatrix} 1 \\ ...
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0answers
17 views

satellites attitude determination TRIAD - how are orbital reference frame vectors constructed?

I posted this same question on space.stackexchange but never received any answer. So I am posting here hoping to get an answer as this is a quite mathematical topic. I am trying to fully understand ...
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1answer
11 views

Function Inequality

Let $E$ and $F$ be normed vector spaces and $\mathscr{L}(E,F) = \{f:E \rightarrow F \mid f$ is linear and continuous$\}$ be a normed vector space with the norm $\lVert f \rVert = \sup_{|x|=1} \{|f(x)| ...
-2
votes
0answers
12 views

Find the dimensions spanned by the vectors [on hold]

Compute the dimension of the subspace spanned by each subset. $\{1, e^{ax}, xe^{ax}\}, \{1, \cos 2x,\sin^2 x\}$
2
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0answers
21 views

Defining a function on a Euclidean Vector Space

Let $\{x, y\}$ be a linearly independent set in a Euclidean space V. Define $f : \mathbb{R} \rightarrow \mathbb{R}$ by $f(a) = \|x − ay\|$ I do not understand the question. How should I define that ...
0
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1answer
28 views

The set $W^{⊥⊥}$ in a Hermitian space

Problem Statement: Let $W$ be a subspace of a Hermitian space $V$. Prove that $W^{⊥⊥}=W$ I am trying to figure out a good strategy for this proof. I know that: $W$ is a subspace of $V$ ...
0
votes
1answer
14 views

problem on annihilators on finite dimensional spaces

Suppose $V$ and $W$ are subspaces of a finite-dimensional vector space $U$. Show that if $V^0 \subset W^0$ then $W \subset V$ This is an exercise problem in Linear Algebra Done Right, 3rd ...
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1answer
30 views

“vector” vs “point” in definition of directional derivative

Given a function $f\colon \mathbb R^n\to\mathbb R$, and given $x,v\in\mathbb R^n$, it is customary to define the "directional derivative of $f$ in the direction $v$ at the point $x$" by $$ D_v f(x) = ...
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2answers
22 views

Direct Sum of $n$ Subspaces

I just need some guidance to prove a portion of the following theorem. Let $V_1, V_2, ... , V_n$ be subspaces of a vector space $V$. Then the following statements are equivalent. $W = \sum ...
1
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1answer
32 views

Find bases for kernel and image of T where $T: P_2 \to M_2$

T is defined as $$T:P_2(\mathbb R) \to M_2 (\mathbb R) \ \text{where} \ T(ax^2 +bx+c)=\begin{pmatrix}-2a +c & b+c\\-3b-3c&6a-3c\\ \end{pmatrix}$$ and I need t find bases for $ Im(T) $ and ...
0
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1answer
14 views

Find the basis when integration is in the condition

Let $V$ be the set of all polynomial $f(x)$ in $P_2$ s.t. $\int_{0}^3 f(x) dx =3f(1) $ If $V$ is a subspace of $P_2$ find a basis of $V$. Can somebody help me get started? The integral condition ...
2
votes
2answers
54 views

Finding whether a vector is in the span of a set of vectors

$$U=\operatorname{span}(v_1,v_2,v_3)$$ where \begin{aligned} v_1=(1,1,1,2) \\ v_2=(1,2,3,1) \\ v_3=(0,1,2,-1)\end{aligned} I need to find if $u=(2,1,0,5) \in U$ If $u \in U$ it is a linear ...
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2answers
20 views

Why do the coefficients of all polynomials of degree at most $d$ as coordinates of a vector in $\mathbb{R}^{d+1}$ lie in ${R}^{d+1}$'s unit sphere?

Consider the coefficients of all polynomials of degree at most $d$ as coordinates of a vector in $\mathbb{R}^{d+1}$. Why would it suffice to suffices to assume that this vector lies in the unit ...
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2answers
248 views

How to prove $\dim(U)=\dim(W)=\dim(V)-1 \implies V=U+W$ based on the following assumption?

Suppose $U$ and $W$ are subspaces of a vector space $V$ such that $\dim(U) =\dim(W)$ and $U\ne W$, how to prove $\dim(U)=\dim(W)=\dim(V)-1 \implies V=U+W$? My approach is to use ...
-1
votes
1answer
41 views

isomorphic linear spaces [on hold]

Let $S$ be the space of $3\times 3$ skew-symmetric real matrices. Then $dim\ S=3$. Is it true that $S$ is a vector isomorphic to $\mathbb R^3$? What is the isomorphism then?
5
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1answer
22 views

Zauner's conjecture

The conjecture is as follow: In $\mathbb{C}^{n}$, there exists $\{v_1,\cdots,v_{n^2}\}$ such that the following holds: $$ \left| \left \langle v_i, v_j \right \rangle \right| = \begin{cases} 1 ...
0
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1answer
43 views

Vectors spaces: $V=U+W$. Technique for showing that an element belongs to $U$?

Let $V$ be finite-dimensional vector space and let $U$, $W$ be subspaces of $V$. Suppose that $V=U+W$. Is there a standard argument for showing that an element belongs to $U$? For example, if the ...
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votes
2answers
49 views

Prove that color space a vector space [closed]

I am trouble in proving that color space is a vector space. Please if anyone knows how to prove this, share your solution. Just consider RGB color model as it can be represented as 3 vectors. Now we ...
2
votes
1answer
49 views

Find the kernel of the linear transformation

So the question asks: find the kernel of the linear transformation $T : \mathbb{R}^4 \to \mathbb{R}^3$ defined by $T(x) = Ax$ where $A$ is the matrix: $$\begin{bmatrix}1 & 0 &1 & 0\\0 ...
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3answers
37 views

Suppose $U$ and $W$ are subspaces of the vector space $V$. Show that $U + W$ = $sp(U \cup W)$

I am not sure where to start with this one, any help would be appreciated. I tried using the definition of a span but I couldn't see where to go from there.
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1answer
17 views

Complementary subspace of $M=(p(2x)=p(x)) ,p\in P_4$

Can anyone please help me with: Find a some base for complementary subspace of $$M=(p\in P_4 : p(2x)=p(x+1)), $$
1
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2answers
48 views

If $V$ is a vector subspace of $R^n$, prove that $V^ \bot$ is a vector subspace of $R^n$

Let $V$ be a subspace of $R^n$. Let $V^ \bot$ be a subset of $R^n$ defined by: $V ^ \bot$ = {$\vec x \in R^n$: $\vec x * \vec v = 0$ for all $\vec v \in V$} Prove that $V^ \bot$ is a subspace of ...
0
votes
2answers
36 views

Why are these not bases in $\mathbb{R}^4$?

I know that bases vectors must span and be linearly independent. The (i) is not bases because the last vector contains $\pi$. The (iii) is not bases because they are not linearly independent. The ...
0
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1answer
21 views

Why is a map to a smaller dimensional space not injective?

I am trying to proof the theorem that states that: "A map to a smaller dimensional space is not injective" So I first suppose that V and W are finite-dimensional vector spaces such that dimV > dimW ...
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4answers
40 views

Prove that $U+W = \{u+w\mid u\in U, w\in W\}$ is a finite-dimensional subspace of $V$

Let $V$ be a vector space over a field $k$ and let $U,W$ be finite-dimensional subspaces of $V$. Prove that $U+W = \{u+w\mid u\in U, w\in W\}$ is a finite-dimensional subspace of $V$. I know how to ...
0
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1answer
53 views

subset $W$ of $\mathbb{R}^3$ question

So the question says: consider the subset $W$ of $\mathbb{R}^3$ consisting of all vectors $$ \begin{bmatrix} x \\ y \\ z \end{bmatrix} \qquad \text{such that } x+y+z \geq -1 $$ Select all statements ...
1
vote
1answer
64 views

Prove that $\dim(U+W) + \dim(U\cap W) = \dim U + \dim W$

Let $V$ be a vector space over a field $k$ and let $U$, $W$ be finite-dimensional subspaces of $V$. Prove that $\dim(U+W) + \dim(U\cap W) = \dim U + \dim W$ I'm given that to begin this ...
2
votes
1answer
15 views

Applying the Dimensional Formula to Prove a Corollary

I am to prove the following corollary. Let $V_1$ and $V_2$ be subspaces of a $n$-dimensional vector space $V$. If the sum of the dimensions of $V_1$ and $V_2$ is greater than $n$, then $V_1$ and ...
0
votes
3answers
54 views

If $\phi(v_1),…\phi(v_\rho)$ are linearly independent, show that $v_1,…,v_\rho$ are linearly independent

Let $\phi:V\rightarrow W$ be linear. Suppose that $v_1,...,v_\rho \in V$ are such that $\phi(v_1),...\phi(v_\rho)$ are linearly independent in $W$. Show that $v_1,...,v_\rho$ are linearly independent. ...
0
votes
1answer
17 views

Idea behind the tangential vector space?

I am currently reading a chapter about Pfaff forms, but not really understand, why the author introduces tangential vector spaces - the definition seems rather redundant to me, if I didn't overlook ...