0
votes
4answers
32 views

Checking subspace

Let $B$ be a fixed matrix in $\mathbb{R}^{n\times n} $ and $W=\{{A \in \mathbb{R}^{n\times n} :AB=BA}\}$ Then is $W$ a subspace of $\mathbb{R}^{n\times n}$ ? I have tried this so far: a) The zero ...
0
votes
2answers
42 views

How to determine whether it is vector space? [on hold]

Does the set of all polynomials of degree exactly $5$, together with all the constant polynomials,determine a vector space?
0
votes
1answer
33 views

My first proof related to subspaces (vector spaces). Please comment.

What do you think about my first proof which deals with subspaces? Theorem An intersection of subspaces is a subspace. Preliminaries Corresponding to the notation in Wikipedia, symbols for vectors ...
0
votes
1answer
18 views

The nullity of a square matrix with linearly dependent rows is at least one. TRUE OR FALSE

Here is the answer my textbook gives. http://imgur.com/ycCRoWK I wonder: Why does the author ask this question specifically for square matrices? Is it different for other matrices.
0
votes
1answer
33 views

TRUE OR FALSE: Matrices with linearly independent row and column vectors are square.

Here is the answer of my textbook: http://imgur.com/vEoY31O Why must a matrice with linearly independent vectors have nullity(A)=0? That is where I lose track of the question. Are zero rows ...
0
votes
1answer
43 views

Can I say that a space of R³ is a subspace of R³?

I am dealing with some questionsm asking if some Set of variables are Vector Spaces of R³. My question is a simple one, a matter of interpretation (I couldn't find a clear answer). Asking if a Set is ...
0
votes
1answer
32 views

Help explain linear algebra/differential calculus theorem in simpler terms.

On a previous question, I got something related to linear algebra and linear algebra, but having no background in linear algebra and a little background in vector calculus(mainly from physics), I ...
0
votes
1answer
19 views

Problem related to dual space of infinite dimensional v.space $V$

Let $V$ be a $K$-infinite dimensional vector space, and let $\mathcal B$ be a basis of $V$. For each $v \in \mathcal B$, let $\phi_v \in V^*$ given by $\phi_v(v)=1$ and $\phi_v(w)=0$, for all $w \in ...
0
votes
1answer
34 views

If $u$ and $v$ are vectors in $3$-space, then $u\cdot v$ is a scalar

My understanding is that B is definitely true because of the below picture but I cannot understand A. Please would someone point me to the right direction! Thanks!
3
votes
1answer
52 views

Sums of special vectors

Let $v$ be a vector obtained by taking a sum of $k$ vectors the of the form $(0,0,\ldots,0, -n, *,*,\ldots,*)$, where $"*"$ stands for either $0$ or $1$, and the position of the $-n$ entry can vary ...
6
votes
2answers
259 views

Linear algebra - Memorising proper definitions of homomorphism types

I am reading a book about linear algebra. On the basis of this book, I worked out the terminology below. Problem: To me, it looks like Wikipedia defines homomorphism differently. Apart from that: Do ...
2
votes
3answers
32 views

Vector spaces - Non-uniqueness of element with property of scalar-multiplicative identity element?

I am dabbling in vector spaces, thinking about the axioms on Wikipedia. Notably, $$1 \mathbf{v} = \mathbf{v},$$ i.e. identity element of scalar multiplication (IEOSM), attracted my attention. I am ...
0
votes
0answers
46 views

prove that $\sum_{k=1}^\infty|x_k y_k|$ converges

Let $V$ be the space of real sequences $x_k$ so that $\sum_{k=1}^\infty x_k^2$ converges. Let $\langle x,y\rangle=\sum_{k=1}^\infty x_k y_k$ Prove that $\sum_{k=1}^\infty |x_k y_k|$ converges My ...
1
vote
1answer
30 views

Maximising a sum with respect to the unit ball.

Suppose that I have a vector $\boldsymbol{v} \in \mathbb{R}^d$, for some dimension $d>1$, and suppose I want to consider the sum $$ \begin{align*} \left(\sum_{k=1}^{d}v_k\right)^2. \end{align*} $$ ...
0
votes
2answers
31 views

Same column space is equivalent to same row space?

If $A$ and $B$ are $n \times n$ matrices that have the same column space, then $A$ and $B$ have the same row space. Can one prove or disprove this? This is my continuation of Same row space is ...
0
votes
1answer
39 views

Same row space is equivalent to same column space?

If $A$ and $B$ are $n \times n$ matrices that have the same row space, then $A$ and $B$ have the same column space. This is false of course. I could just come up with examples though. Can one prove ...
1
vote
2answers
25 views

Vector Spaces: canonical basis for the usual vector spaces

I'm looking for a listing of the canonical basis for the most common vector spaces. Some standard basis are not so obvious. For instance, the basis for vector space $\Bbb C^2$ is $\{ ...
0
votes
1answer
13 views

Prove the number of unordered pairs of linearly independent elements

Let $V$ be a vector space over $K$. Let $K={\mathbb{Z}}/{p\mathbb{Z}}$, and $\dim V=3$. We know that $V$ has $p^3$ elements. I need to show that the number of unordered pairs of linearly ...
0
votes
3answers
75 views

High dimensional vector space references

Is there any good text book or review papers that introduce high dimensional vector spaces and its peculiarities as compared to generic/low-dimensional vector spaces? For example, high dimensional ...
-1
votes
0answers
22 views

Positive definite [closed]

I need a graphical representation of positive definite from the eigen values of the matrix which can be expressed from the second degree homogenous equation.
0
votes
1answer
60 views

Quaternion expansion

I have a quaternion equation $ \psi(s)=Pe^{\frac{1}{2}k(s)}\tag 1$ Given conditions and data Here P is a constant unit Quaternion defined for 3D rotation matrix as $(p_1,p_2,p_3,p_4) , p_4\in ...
0
votes
0answers
37 views

Positive definite matrix. [closed]

How to illustrate the positive definite matrix in vector space by using the eigen values and eigen vectors?
0
votes
2answers
77 views

Show that vectors of the form $(a,b,1)$ do not form a vector space

Show that vectors of the form $(a,b,1)$ do not form a vector space I tried using the vector space axioms to attack the problem but I am not going anywhere with this problem. I do not need a ...
2
votes
1answer
38 views

Vector spaces - Multiplying by zero vector yields zero vector.

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
1answer
49 views

A subset that is closed under multiplication but not addition? [duplicate]

I can't get my head around subspaces despite having studied on them quite a lot. Here goes: The problem statement, all given variables and data Give an example of a non-empty subset U of R^2 such ...
1
vote
0answers
36 views

Vector spaces - Multiplying by $-1$ yields inverse element of vector addition.

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. The following proof is based on vector space related axioms. Axiom ...
0
votes
1answer
25 views

Subspace of a vector space Definition

If $W$ is a subspace of a finite-dimensional vector space $V$, then: $\dim(W) \leq \dim(V)$. That makes me think about the definition of a subspace. For example, in $\Bbb R^3$, is $\Bbb R^3$ ...
5
votes
1answer
79 views

Must a normed vector space be over $\mathbb{R}$ or $\mathbb{C}$?

If it must be, why is this so? In the maths courses I have taken normed vector spaces always have been over $\mathbb{R}$ or $\mathbb{C}$, but I don't see that this has to be so. I am asking because I ...
0
votes
1answer
24 views

linear map mapping linearly independent sets [closed]

Let $V_1,V_2$ be finite-dimensional vector spaces. $f: V_1\to V_2$ be a linear map. Suppose for $x_1,x_2,\ldots, x_k\in V_1$ we get $\{f(x_1),\ldots, f(x_k)\}$ is a linearly independent set. Does it ...
2
votes
1answer
43 views

Existence of a subspace with a certain property

I am having trouble solving this problem.I have started solving the problem , so far my guesses for the subspace U were the intersection of V and complement of KerT , but i was soon able to come up ...
0
votes
0answers
24 views

Showing the conjugate symmetric property of an inner product when we don't know if our field is $\mathbb{C}$.

The conjugate symmetric property of an inner product states that $\langle{x, y}\rangle = \overline{\langle{y, x}\rangle}$. My question is regarding showing this when we don't necessarily know that our ...
1
vote
2answers
17 views

Dimension of an intersection.

I have a problem that if $U$ and $V$ are two subspaces of $\mathbb{R}^n$ and $\dim(U)>n/2$, $\dim(V)>n/2$, show that $\dim(U\cap V)\geq 1$. Now I know that $$\dim(U\cap ...
0
votes
0answers
15 views

Show that U subspace is supplementary to the kernel. How to find values of a b c d using intersection of two matrices.

I already found the kernel to be \begin{pmatrix} -2c&-2d\\c&d \end{pmatrix}. and U is a subspace of a $M_2$ matrix defined by \begin{pmatrix} a&b\\2a&2b \end{pmatrix}. So i have to ...
3
votes
2answers
38 views

Infinite-dimensional space

I've been dealing with an exercise asking to show that the infinite-dimensional space $R^\infty$ of infinite sequences is isomorphic to a proper subspace of itself. At first I thought I had to show ...
0
votes
2answers
48 views

Space of matrices that commute with a given matrix

Let $A$ be an $n\times n$ complex matrix, and $C(A)$ be the vector space of all matrices that commute with $A$. I have to determinate if the dimension of $C(A)$ is greater or equal than $n$, or not. ...
0
votes
1answer
34 views

Doubt about subspaces being vector space

Whenever i am saying $V$ is a $n$-dimensional vector space, it means it has $n$ basis vectors each with n elements, right. So when i am proving some theorems or relations involving some ...
0
votes
1answer
11 views

Nonhomogeneous Systems of m equations in n unknowns and Solution Spaces.

My book says that solutions sets of nonhomogeneous systems of m equations in n unknowns is NEVER a subspace of R^n. Why? If we look at any two planes intersecting in R3, there may be a line formed. ...
1
vote
0answers
21 views

Acute angle between plane and line

Find the acute angle between: $x-y-3z=5$ and $x=2-t$ $y=2t$ $z=3t-1$ Here is how I proceed. I take the dot-product of the normal of the plane and the directional vector of the line. This gives me ...
0
votes
3answers
39 views

Show that a linear matrix transformation is bijective iff A is invertible.

Suppose a linear transformation $T: M_n(K) \rightarrow M_n(K)$ defined by $T(M) = A M$ for $M \in M_n(K)$. Show that it is bijective IFF $A$ is invertible. I was thinking then that I could show ...
1
vote
0answers
15 views

Proving that $b=(e_1,…,en_,f(e_1),…,f(e_n),f^2(e_1),…,f^2(e_n))$ is base of $E$

Suppose that $E=S\oplus \ker{f^2}$, and $f$ an endomorphism, $S$ is the base of $(e_1,...,e_n)$ and $f^3=0_{L(E)}$. How can I prove that ...
1
vote
2answers
28 views

Do polynomials $ P(t)$ of an odd degree have at least one real root belong to $(t-a)Q(t)$?

This is a continuation of a question where ker(T) = (t-a)Q(t) = P(t). Show that {P(t) ∈ R[t] | deg(P(t)) = 3} ⊂ $∪_{a∈R}$ker(T). So the mark scheme says that all polynomials in R[t] of an odd ...
3
votes
0answers
64 views

Vector spaces - Multiplying by zero scalar yields zero vector

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 related axioms. ...
1
vote
0answers
57 views

What is the solution of greatest possible dimension for this decoupled linear subspace problem?

Let $\left\{A_i\right\}$ be a $k$-element set of $n\times n$ Hermitian matrices, and let $P$ be an $n\times n$ rank-$m$ orthogonal projection matrix. We consider the projection of any matrix $A$ onto ...
4
votes
1answer
47 views

Vector spaces - If an addend adds nothing, then the addend is the zero vector.

Please rate and comment. I want to improve; constructive criticism is highly appreciated. Please take style into account as well. With one exception, the following proof is solely based on vector ...
1
vote
0answers
43 views

Linear algebra proves

Hi does anyone know how to prove or disprove the following statement. If $V$ and $W$ are subspaces of $R^n$ and $v\in V$ and $w \in W$ then $v + w \in V ∪ W$.
0
votes
0answers
10 views

For what values of λ is this family free (independent), spanning and a basis of R[t]≤3

The family of polynomials $F$ = {${(λ^2 − 1)t^3 + t^2, λt^3 + t − λ, (1 − λ)t^3 + t + 1, λ}$} in $R[t]_{≤3}$ I set their sum to 0 to find the values for it to be independent. $a((λ^2 − 1)t^3 + t^2) ...
0
votes
1answer
27 views

Metric for vector sets

I am currently working on a classification algorithm. Each class is represented by a set of 3D vectors. The cardinality differs for each class. The order of the vectors in a set is completly random. ...
1
vote
0answers
26 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 ...
3
votes
5answers
311 views

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. Proof of uniqueness of identity element of addition of vector space ...
1
vote
1answer
36 views

Trouble proving that $\dim{(W1+W2)}=\dim{(W1)}+\dim{(W2)}-\dim{(W1\cap W2)}$

Let me first summarize the part I understand: $\dim{W_1}=n$, $\dim{W_2}=m$, $\dim{(W_1\cap W_2)}=k$ I've defined $C$ as a basis for $W_1\cap W_2$ with $C = \{v_1, ... v_k\}$. I've expanded $C$ ...