2
votes
3answers
28 views

Prove that if $\dim X'<\infty$ then $\dim X<\infty$

I have to prove that $\dim X'<\infty$ then $\dim X<\infty$ where $X$ is a normed vector space and $X'$ is a space of all linear and continuous functionals from $X$. How can I prove this? I ...
0
votes
3answers
43 views

Show that if the set $\{v_1,v_2,v_3\}$ is linearly independent then so are all subsets. [on hold]

Show that if the set $\{v_1, v_2, v_3\}$ is linearly independent then so are $\{v_1, v_2\}$, $\{v_1,v_3\}$, $\{v_2,v_3\}$, $\{v_1\}$, $\{v_2\}$ and $\{v_3\}$. I don't even know what to start with. Do ...
0
votes
2answers
29 views

Proof that if we add a vector to a linearly dependent set of vectors in a vector space $V$, then the new set of vectors is still linearly dependent

Prove that if $S=\{v_1, v_2, v_3\}$ is a linearly dependent set of vectors in a vector space $V$, and $v_4$ is any vector in $V$ that is not in $S$, then $\{v_1, v_2, v_3, v_4\}$ is also linearly ...
1
vote
2answers
46 views

Find dimension of ℒ $(V)$ and polynomial that brings every linear transformation to $0$

Here's the prompt: Let V be a vector space of finite dimensions $n$ over the field $\mathbb{F}$, and let $\tau \in$ ℒ $(V)$. What is the dimension of ℒ $(V)$ as a vector space over $\mathbb{F}$? With ...
1
vote
1answer
17 views

Find a vector $t \in \{x,y,z\}$ with base $\{u, v, w\}$

I don't know how to find a vector $\vec t$ that will suffice the condition: $\vec t \in \{x,y,z\}$ with bases $\{u, v, w\}$ the given vectors are: $$ \begin{array}{rcrrrrrl} u &=& [ & ...
-1
votes
0answers
13 views

Show that a plane in R^3 is a vector space over R if and only if the origin lies in the plane.

Show that a plane in $\mathbb{R}^3$ together with the usual addition and multiplication is a vector space over $\mathbb{R}$ if and only if the origin lies in the plane. (Hint: you may use that ...
0
votes
1answer
37 views

Why this set is not a vector space?

Let V =R^2 and define addition and scalar multiplication operation as follows : $u=(u_1, u_2)$ $v=(v_1, v_2)$ $$u+v=(u_1+v_1,u_2+v_2)$$ $$ku=(u_1k,0)$$ The book says : "the addition operation is ...
0
votes
1answer
24 views

Why $\dim(\hom(V,W))=\dim(V) * \dim(W)$?

I have found that the question I want to ask someone had asked, here is the website: $\hom(V,W)$ is canonic isomorph to $\hom(W^*, V^*)$ Here is my question: Why $\dim(\hom(V,W))=\dim V * \dim W $? ...
2
votes
1answer
28 views

What is meant by $\langle \cdot,\cdot \rangle ^H_\mathbb{R}$?

there is the following statement: Let $\langle \cdot,\cdot \rangle_\mathbb{R} = \sum_{k = 1}^{n} x_k y_k$ be that standard Euclidian scalar product in $\mathbb{R}^n$ and $\langle \cdot,\cdot ...
2
votes
2answers
63 views

On the canonical isomorphism between $V$ and $V^{**}$

I am trying to understand more about the Bidualspace (or double dual space). The whole idea is that $V$ and $V^{**}$ are canonically isomorphic to one another, which means that they are isomorphic ...
1
vote
1answer
43 views

Prove that the mapping $\psi : L(V,W) \rightarrow L(W^*, V^*)$ given by $\psi(T) = T^t$ is an isomorphism.

Let $V,W$ be finite-dimensional vector spaces over the same field $\mathbb{F}$ and let $L(V,W)$ be the vector space of $\mathbb{F}$-linear transformations from $V$ to $W$. Prove that the mapping ...
1
vote
2answers
16 views

concept between group and vector space, compare G/N with V/W

When we considered factor groups G/N, we need N to be normal,but in vector space V/W, why W only be subspace?
1
vote
2answers
19 views

Determine if all vectors of the form (a,b,c), where b=a+c+1 are subspaces of R^3?

Determine if all vectors of the form $(a,b,c)$, where $b=a+c+1$ are subspaces of $\mathbb{R}^3$? Use the theorem: If $W$ is a set of one or more vectors from a vector space $V$, then $W$ is a ...
0
votes
1answer
57 views

inner product space , dual space, proof about isomorphism

Let $V$ be a vector space (not necessary being finite dimensional) and let $U,W$ be subspaces of $V$ such that $V = U\oplus W$. Prove that $V^\ast/(W^0)$ is isomorphic to $W^\ast$. Notation and ...
0
votes
1answer
33 views

V = U⊕W then Prove that (V/W)* is isomorphic to W^0

Let $V$ be a vector space (not necessary being finite dimensional) and let $U$, $W$ be subspaces of $V$ such that $V = U\oplus W$. Prove that $(V/W)^*$ is isomorphic to $W^0$. note: (V/W)* is the ...
0
votes
1answer
22 views

Prove Linear Dependence in T: V -> W

Problem: "Let $V$ and $W$ be vector spaces and let $T:V \rightarrow W$ be a linear transformation. Prove that, if $\{v_1, v_2, v_3\}$ is a set of three linearly dependent vectors in $V$, then the set ...
0
votes
0answers
25 views

Why is $\hat{x}$ in the linear regression equation $A^TA\hat{x} = A^Tb$ part of $C(A^T)$

When finding the best fit line for a number of points, we use $A^TA\hat{x} = A^Tb$ where we solve for $\hat{x}$. I understand that the projection $p=A\hat{x}$ is part of the column-space of $A$ and ...
1
vote
3answers
69 views

An infinite generating set of a finite dimensional vector space contains a basis

Let $S$ be an infinite generating set of a finite dimensional vector space , then how do we prove that there is a subset of $S$ which is a basis of the vector space ? Please help
1
vote
2answers
17 views

Finding an Orthonormal Basis using Gram Schmidt

Given the set of vectors $S=${${V_1=\binom{1}{4},V_2=\binom{4}{-4} }$} I am to find an orthonormal basis for $R^2$ using the Gram-Schmidt process. I've already worked it out and found the orthonormal ...
2
votes
1answer
19 views

Defining operations for a vector space

I was hoping someone could help me with the following. Is it possible to define operations + and $\cdot$ on this set to make it a vector space: \begin{equation*} ...
0
votes
1answer
19 views

The difference between norm and modulus

I'd like to know the difference between norm of a vector, ||v|| and the modulus of a vector, |v|
3
votes
2answers
41 views

Arbitrary (i.e. not necessarily finite-dimensional) vector spaces; reference request.

Its virtually impossible to complete an undergraduate degree these days without studying finite-dimensional vector spaces in quite some detail. So like most of us, I've done all that; however, just ...
1
vote
3answers
29 views

A question on basis of vectorspaces and subspaces

Let $V$ be a finite dimensional vector space and $W$ be any subspace . It is known that if $A$ is any basis of $W$ then by "extension-theorem" , there is a basis $A'$ of $V$ such that $A \subseteq ...
1
vote
1answer
34 views

Determining and enforcing linear dependence

Assuming we have a large set of multi-dimensional vectors (20k vectors, 100 dimensions each). My questions are the following: How can we determine the level of linear dependence of this set? Is ...
0
votes
0answers
11 views

Transformation Matrix of a function

I have the following: (Note: $V^{*}$ is defined as: $V^{*} = \{ L: V \rightarrow \mathbb{R} | \text{L is linear} \}$) Let $V$ be an $\mathbb{R}$-Vectorspace. Let $\phi \in V^{*} \text{ \ } \{0 \}$ ...
1
vote
1answer
32 views

What is the canonical basis of a dualspace in $\mathbb{R}^3$?

I have the following: Consider the basis $$B := \{\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} -1 \\ 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 2 \\ 2 \\ 1 \end{pmatrix} \}$$ of the ...
1
vote
1answer
25 views

Why we take transpose of Vector (Displacement Vector)?

I'm trying to understand some equations that involves transpose of vectors (displacement vectors to be precise) Two set of vectors F and G (with i,j) that corresponds to X,Y value in plane and ...
0
votes
1answer
57 views

Finding the dimension of subspace span(S)

Problem: Consider the set of vectors $S= \{a_1,a_2,a_3,a_4\}$ where $a_1= (6,4,1,-1,2)$ $a_2 = (1,0,2,3,-4)$ $a_3= (1,4,-9,-16,22)$ $a_4= (7,1,0,-1,3)$ Find the dimension of the subspace $span(S)$? ...
0
votes
0answers
41 views

Calculating the null space of a matrix [closed]

I am sorry for maybe this is a duplicate question but I really need someone to help me with this I am trying to calculate the null space of this matrix, but I really don't know how and I also have ...
-1
votes
1answer
34 views

Uinviersal property of basis of a vector space

Let V be a vector space over a field k. Let B be a subset of V. If any set map from B to any vector space W can be extended uniquely to a k-linear map from V to W. Then B is a basis of V. Can ...
0
votes
1answer
11 views

what is the dimension of this subspace for given problem

In a subspace $W=\{[a_{ij}]:a_{ij}=0$ if $i$ is even$\}$ of all $10\times 10$ real matrix, what is the dimension of W?
0
votes
1answer
26 views

are there any unordered basis,what is the most basic example???

I have been doing linear algebra and I can't really understand the existence of basis other than ordered basis ,but since ordered basis are taught as special arrangement basis then what are other ...
3
votes
0answers
32 views

Given three points in $\mathbb R^3$ that define a plane. Need to find the normal of the plane.

I came across this question and it has been troubling me for a while... A plane in $\mathbb R^3$ that contains three points is defined as $A=(1, 2, 3)$, $B=(0, 1, 4)$, $C=(2, 1, -7)$... I have to find ...
4
votes
1answer
85 views

An Algebraic Version of vector spaces

Consider the following set of real numbers $\mathcal{X}=\{1,2,3,\sqrt{2}+1,\pi+\sqrt{2}\}$. Lets consider the set of all linear combinations with integer coefficients of these numbers which I will ...
0
votes
0answers
29 views

Finding the distance between two vectors

If given the basis of a row space is $\{(1, 0, 1, 1), (0, 1, -1, 1)\}$, how can I use this information to find the vector in this row space that is closest to the vector $(1, 1, -1, 1)$? Please ...
0
votes
1answer
26 views

$A \oplus C = B \oplus C$ but $A\neq B$

Let $V$ be a vector space, with subspaces $A, B,C$ such that $A\oplus C = B\oplus C = V$. Prove or give a counterexample disproving that $A=B$. I am trying to find a counterexample.
1
vote
0answers
17 views

Advice on vector equation geometry

I was wondering what a good approach and your technique for learning geometrically what objects are from their vector equations, often involving dot and cross products, are and 'easily' identifying ...
1
vote
1answer
18 views

Trouble understanding finite vector spaces and Gaussian coefficent

I have studied linear algebra for 2 months now and i cannot understand a task that i am currently trying to solve. Basically i am trying to find the amount of bases for n-dimensional vector space over ...
0
votes
1answer
40 views

Finding basis of vector spaces

Without proof find the dimension and a basis of the following vector spaces $V$ over the given field $K$. $V$ is the set of all polynomials over $\mathbb{R}$ of degree at most $n$, in which the sum of ...
3
votes
3answers
54 views

Regarding a Basis for Infinite Dimensional Vector Spaces

In my linear algebra class, during the discussion of vector spaces, our instructor mentioned infinite dimensional spaces, including the polynomial space over Q and the space of all continuous ...
1
vote
1answer
19 views

sum of two matrices question given condition

How can it be proved that two matrices being orthogonally diagonalizable indicates that their sum is also?
0
votes
1answer
29 views

Explain why each set is NOT a basis for the given vector space

My biggest problem with linear algebra is trying to get the wording right when I answer questions. I want to communicate my answers as effectively as possible. So here are my answers to the following ...
1
vote
0answers
27 views

three dimensional subspace question

If a vector is in $\mathbb{R}^5$, does this mean that the projection of this vector onto $S$ is in $\mathbb{R}^3$, where $S$ is some 3-dim subspace of $\mathbb{R}^5$?
0
votes
1answer
24 views

Inner product over the $C^2$

Let a, b, c, d ∈ C and consider the vector space $C^2$ Suppose inner product is defined as: $⟨x, y⟩ = ax_1\bar y_1 + bx_2\bar y_1 + cx_1\bar y_2 + dx_2\bar y_2$ I am trying to find all a, b, ...
0
votes
1answer
39 views

Find a basis of the $k$ vector space $k(x)$

Suppose $x$ is a transcendental over field $k$ and $k(x)$ is the field of fractions of $k[x]$. Can we explicitly express a basis of the $k$ vector space $k(x)$?
0
votes
2answers
43 views

General Vector Space: Change of basis

If $P$ is the transition matrix from a basis $B'$ to a basis $B$, and $Q$ is the transition matrix from $B$ to a basis $C$, what is the transition matrix from $B'$ to $C$? What is the transition ...
0
votes
1answer
39 views

find the number of solutions of the equation $a_1x_1+a_2x_2+…+a_nx_n=0$ in a linear space over Galois field

Linear space $\Bbb F_p^n$ contains $p^n$ vectors $( x_1, x_2, ..., x_n)$ with length $n$ over finite $\Bbb F_p$ Galois field comprised from $p$ elements. How many solutions in $\Bbb F_p^n$ has the ...
1
vote
0answers
28 views

Finding a basis for span of vectors

U = span{(1,0,0),(0,2,-1)}. W = span{(0,1,-1)} How can I find bases for U and W ? (I think they're linearly independent, right?) can I just take B1 = {(1,0,0),(0,2,-1)} for U, and B2 = {(0,1,-1)} for ...
0
votes
4answers
61 views

Help understanding the span of a set in $R^2$

I feel so lost in this section regarding vector spaces, sub spaces, spanning sets and basis. I understand the basic concepts regarding vectors where basically a vector gives magnitude and direction. ...
1
vote
2answers
46 views

Consistency of matrix norm: $||Ax||_2 \leq ||A||_{Frobenius}||x||_2$

I'm trying to show that $||Ax||_2 \leq ||A||_{F}||x||_2$ where $A$ is an n by n matrix, $x\in \mathbb R^n$, $||x||_2$ is the euclidean norm, and $||A||_F$ is the frobenius norm. I actually wrote ...