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
29 views

If I want to find the dimension of the image of a linear transformation…

If I have a linear transformation $T(v)=Av$ and want to find the dimension of the range$(T)$, the following procedure is valid? Looking at the columns of $A$, if all columns are linearly independent, ...
0
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3answers
98 views

Finding eigenvalues of $A^{10} + A^7 + 5A$.

Problem: Let $A = \begin{pmatrix} 1 & 2 & -1 \\ 0 & 5 & -2 \\ 0 & 6 & -2 \end{pmatrix}$. 1) Compute the eigenvalues of $A^{10} + A^7 + 5A$. 2) Compute $A^{10} X$ for the ...
0
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3answers
111 views

Prove that $\mathbb{R}^∞$ is infinite-dimensional.

Prove that $\mathbb{R}^∞$ is infinite-dimensional. The section that contains this problem deals with the idea of a basis, so the proof probably has something to do with it (since a basis must have a ...
3
votes
2answers
102 views

Is this matrix diagonalizable over $\mathbb{R}$ or $\mathbb{C}$?

Problem: Let $A = \begin{pmatrix} 6 & 0 \\ -2 & 2 \end{pmatrix}$. Is this matrix diagonalizable over $\mathbb{R}$? If not, is it diagonalizable over $\mathbb{C}$? Compute the eigenvalues $\...
1
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3answers
88 views

Find eigenvalues and eigenvectors of this matrix

Problem: Let \begin{align*} A = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 \end{pmatrix}. \end{align*} Compute all ...
1
vote
1answer
66 views

In an icosahedron subdivided n times, how can I find the coordinates of adjacent centroids?

I think it would be helpful to refer to this image when trying to follow my description: http://i.imgur.com/nRXQo3W.jpg (taken from http://experilous.com/1/blog/post/procedural-planet-generation). ...
0
votes
1answer
41 views

Vector spaces - $\mathbb{R}^n$ and $\mathbb{R}^m$

I stumbled on the following text on Wikipedia: Suppose the random column vectors X, Y live in $\mathbb{R}^n$ and $\mathbb{R}^m$ respectively, and the vector $(X, Y)$ in $\mathbb{R}^{n+m}$ has a ...
3
votes
3answers
146 views

Proving the first part of the diagonalization theorem - eigenvalues and eigenvectors

Let $d(\lambda_i)$ represent the geometric multiplicity, and let $m(\lambda_i)$ represent the algebraic multiplicity. Theorem: Let $T$ be a linear operator on an $n$-dimensional vectorspace. Then $T$ ...
1
vote
1answer
45 views

Finding a vector orthogonal to columns of a matrix.

Given a matrix $X$ of sixe $n\times m$ with $m>n$ or $m<n$ how to find a vector orthogonal to all the $m$ columns of $X$ in the most computationally efficiemt way.
4
votes
5answers
491 views

Unable to understand the proof of two isomorphic finite-dimensional vector spaces having the same dimension

Theorem: Two finite-dimensional vector spaces are isomorphic if and only if they have the same dimension. I can understand how to prove that if they are isomorphic then they have the same dimension. ...
8
votes
2answers
664 views

What makes a vector an object with both magnitude and direction?

According to my understanding, A vector is an element of a set called the vector space which satisfies a list of axioms like : closure under vector addition, closure under scalar multiplication, ...
2
votes
0answers
82 views

Fastest way to find linearly independent columns of a matrix

Given a rectangular matrix $X$ of size $n\times m$ with $m>n$, what is the fastest way to find the linearly independent coloums. Robust methods like SVD or RRQR decompostion have complexity of ...
0
votes
1answer
57 views

Regarding subspace of Hilbert space

Suppose I have Hilbert space $H$, and two subspaces $V_1$ and $V_2$. Somehow I know that $V_1 \cup V_2=H$ and $V_1 \cap V_2 =\{0\}$. I want to show that one of them must be trivial. I could think of ...
6
votes
1answer
564 views

Can set of integers form a vector space over field of rationals?

As field of reals $\mathbb{R}$ can be made a vector space over field of complex numbers $\mathbb{C}$ but not in the usual way. In the same way can we make the ring of integers $\mathbb{Z}$ as a ...
1
vote
2answers
112 views

Question from self-studying Halmos' Finite Dimensional Vector Spaces

For section 1 on Fields, there is a question 2c: 2. a) Is the set of all positive integers a field? b) What about the set of all integers? c) Can the answers to both these question be changed by ...
0
votes
2answers
52 views

Finding Linear independent vectors

Thanks for clarifications. Now i am posting the question in a different way. Suppose a vector $V$ is orthogonal to vectors $X1$ and $X2$. $X1$ and $X2$ are linearly independent. Now if $V$ is also ...
1
vote
3answers
87 views

Let $H$ be a non-zero subspace of $V$, and let $T(H)$ be the set of images of vectors in $H$. Prove that $\dim(T(H))\leq \dim(H)$.

Let $V$ and $W$ be finite-dimensional vector spaces and $T$ be a linear transformation $T:V\to W$. Let $H$ be a non-zero subspace of $V$, and let $T(H)$ be the set of images of vectors in $H$. ...
0
votes
1answer
85 views

Determining if a basis consists of eigenvectors

Problem: Let $V = P_1(R)$ be a vectorspace, and let $T(a+bx) = (6a - 6b) + (12a-11b)x$ be a linear operator on $V$. Let $\beta = \left\{ 3+4x, 2+3x \right\}$ be a basis for $V$. Compute $[T]_{\beta}$ ...
0
votes
3answers
78 views

In a vector space in finite dimension, all vectors which are not colinear, are orthogonal. True or false?

This is a theorem I learn few month ago, and I found it funny. Well I found on facebook, some groups about mathematics, and just to test people I said : In a vector space in finite dimension, all ...
1
vote
2answers
60 views

Is this set a vector space?

"The Set of all continuous functions on the interval $[0,1]$" How do I determine this? Do I think of possible functions which have outputs between $0$ and $1$, and determine if the inputs are real ...
1
vote
2answers
73 views

Linear mapping matrix with paramters.

I solved a linear mapping problem recently and it turns out no to be correct, although i thought it was a simple problem. The problem asks me to find real parameters $a,b,c$ such that linear mapping ...
1
vote
1answer
43 views

Finding bases such that the matrix representation is a block matrix where one submatrix is the identity matrix

Problem: Let $L: \mathbb{R}^2 \rightarrow \mathbb{R}^3$ be a linear map with \begin{align*} [L]_{\alpha}^{\beta} = \begin{pmatrix} 2 & 3 \\ 4 & 6 \\ 6 & 9 \end{pmatrix} \end{align*} as the ...
1
vote
1answer
45 views

How to construct an inverse linear map?

Problem: Consider the linear map $L: \mathbb{R}^3 \rightarrow \mathbb{R}^3: (x,y,z) \mapsto (2x+y-z, y-2z, -2x-z)$. Let $U = \text{span} \left\{(0,0,1), (1,1,1)\right\} \subset \mathbb{R}^3$ be a ...
3
votes
2answers
275 views

What is the bar symbol over a complex scalar in the expression $\overline{\lambda}$?

I have the following problem from section 1.4 (Vector Spaces) of Peter Peterson's Linear Algebra textbook. I am having trouble with the way multiplication is defined on the given vector space, $\bf{V}^...
-1
votes
2answers
47 views

Proving that the matrix of a linear transformation with respect to two bases has a particular form

I'm doing the conceptual exercises from my linear algebra book, and I ran up to the following exercise: Let $\mathbb{V}$ be a vector space with basis $\mathcal{B} = \{ \mathbf{v}_1, \ldots , \...
0
votes
2answers
35 views

Determining the coordinates of a vector with respect to a basis

Problem: Let $V = \mathbb{R}[X]_{\leq 4}$ be the vectorspace of all polynomials of degree at most $n$, and let $\alpha = \left\{1, 1 +x, (1+x)^2, (1+x)^3, (1+x)^4\right\}$ be a basis for $V$. Find the ...
1
vote
1answer
29 views

Finding a vector in $\mathbb{R}^2$ given its coordinates with respect to a given basis

Consider the basis $B$ of $\mathbb{R}^2$ consisting of vectors $\begin{bmatrix}3 \\ -5 \end{bmatrix}$ and $\begin{bmatrix} 2 \\ -5 \end{bmatrix}$. Find $x$ in $\mathbb{R}^2$ whose vector relative to ...
4
votes
0answers
133 views

The topology of $GL(V)$

Let $V$ be a topological vector space (not necessarily finite-dimensional) over a field $K$, and let $GL(V)$ be the group of invertible linear maps $V\to V$ under composition. There are two obvious ...
4
votes
1answer
38 views

Show that the number of points of $V(I)$ is at most $m_1m_2…m_n$ if $x_i^{m_i}\in \left\langle \text{LT}(I) \right\rangle$.

This is an exercise from Ideals, Varieties and Algorithms by Cox et al. Let $I\subset \mathbb{C}[x_1,...,x_n]$ be an ideal such that for each $i$, some power $x_i^{m_i}\in \left\langle \text{LT}(...
0
votes
0answers
422 views

Is the direct sum of two orthogonal subspaces well defined in infinite-dimensional vector spaces?

Let's say that $V$ is an inner product space on some field $\Bbb{K}$ and $M$ is a subspace of $V$. If $M^{\perp}$ is the orthogonal complement of $M$ with respect to the inner product, can I make the ...
0
votes
3answers
90 views

Find a basis of a subspace defined by a linear equation

Let $B=\{v_1,v_2,v_3,v_4\}$ be a basis of $V$. Let $$V \supset S= \left \{v:v=\sum\limits_{i=1}^4 \alpha_iv_i, \alpha_1+2\alpha_2+\alpha_3-\alpha_4=0 \right \}$$ Find a basis of $S$. I don't ...
1
vote
3answers
119 views

How to prove there exists a unique linear map such that $T(e_i) = w_i$ in an infinite-dimensional vector space?

Problem: (a) Let $V$ and $W$ be two finite dimensional vectorspaces over a field $F$, and let $\left\{e_1, e_2, \ldots, e_n\right\}$ be a basis for $V$. Then there exists for each $w_i \in W$ an ...
0
votes
1answer
94 views

Prove that an $n$-dimensional non-unital algebra over a field $k$ is isomorphic to a subalgebra $\mathfrak{M}_n(k)$

Assume we have already proved this for unital algebras. Here's my book's solution: Construct the unital algebra $A^1$ [with unit $(1,0)$] as an algebra on the vector space $k\oplus A$ with the ...
0
votes
3answers
39 views

Prove {$v_1,v_2,w$} is a basis for vector space _V_

A problem from my textbook states: Let {$v_1,v_2,v_3$} be a basis for a vector space $V$. Prove that, if $w$ is not in $sp(v_1,v_2)$, then {$v_1,v_2,w$} is also a basis for $V$. Assume {$v_1,v_2,...
2
votes
4answers
70 views

If $V$ is a vector space, then, proving that…

I have a big problem with this problem... : If $V_m(\mathbb{R})$ is a vector space whose dimension is "$m$" then Proving that "$m$" is even number if and only if exist an endomorphism $J$ of $V_m(\...
1
vote
4answers
168 views

Is the given subset a subspace of the given vector space?

The set of all polynomials of degree greater than 3 together with the zero polynomial in the vector space P of all polynomials with coefficients in $\Bbb R$. Let $S$ be the set of all polynomials ...
1
vote
2answers
301 views

Distance between point and plane - why use the dot product?

So according to this, the signed distance between a point and a plane will be the dot product of the plane's normal vector (does it have to be a unit vector?) and the point-in-plane minus the point ...
2
votes
5answers
332 views

Find the dimension of a vector subspace

I'm doing a problem on finding the dimension of a linear subspace, more specifically if $\:$ {$f \in \mathcal P_n(\mathbf F): f(1)=0, f'(2)=0$} is a subspace of $P_n$, what is this dimension of ...
-1
votes
1answer
41 views

All surfaces through a common “concur-line” [closed]

Find all second degree surfaces passing through a common given parameterized space curve of intersection: $$ (x,y,z) = (\pm \sqrt {2 t ( 1-t)} , t , (1-t) ) $$ using a single variable parameter ...
0
votes
1answer
111 views

What is “Real coordinate space”?

What is the Real Coordinate Space in the discussion of vectors? How does it relate to Cartesian Coordinate System and Euclidean Space? P.S. Please, use naive terms.
1
vote
1answer
60 views

Consequences of the positivity condition $v^t A v > 0$ for the eigenvalues of $A$

Let $A$ be an $n \times n$ symmetric real matrix with n distinct eigenvalues $\lambda_1 , ... , \lambda_n$. a) Suppose $v^t(Av)$>0 for all v in $R^n$, v$\ne$0. Show that all $\lambda_i$ are positive ...
2
votes
3answers
185 views

Must a basis for an $n$-dimensional vector space have $n$ vectors?

Does a basis for an $n$-dimensional vector space have to have $n$ vectors? For example, if I form a basis for $\mathbb{R}^n$, do I need at least $n$ vectors in my basis set? In other words, can I ...
0
votes
1answer
205 views

Finding change of basis matrix when given two bases as a set of matrices

Find the change of basis matrix between the following bases: $\alpha = \left\{ \begin{pmatrix} 1 & 1 \\ -1 & 2 \end{pmatrix}, \begin{pmatrix} 3 & 1 \\ -1 & 2 \end{pmatrix}, \begin{...
1
vote
1answer
95 views

Generating vectors in a non-orthogonal 3D lattice with increasing magnitude

I am trying to build an algorithm to generate a sequence of lattice vectors $\mathbf{v}_n$ in 3D such that: (a) the first vector $|\mathbf{v}_1|$ is the shortest vector of the lattice (b) for all $i ...
2
votes
3answers
27 views

Determining the formula for a linear map

Determine the formula for the following linear map: $L : \mathbb{R}^2 \rightarrow \mathbb{R}^2$ with $L(1,2) = (0,-1)$ and $L(-1,-1) = (2,1)$. Attempt at solution: On the basis of these examples I ...
0
votes
1answer
33 views

Sesquilinear Forms: Polarization

This thread is only Q&A.* Given a Hilbert space $\mathcal{H}$. Consider the transforms: $$q[\varphi]:=s(\varphi,\varphi)\quad s(\varphi,\psi):=\frac{1}{4}\sum_{\alpha=0\ldots3}i^\alpha q[\varphi+...
0
votes
0answers
22 views

Linear (In)dependence and other relations

(i) "nontrivial solution" same as "linear dependence" same as "determinant zero" same as "the vectors lie in the same plane". (ii) "trivial solution" same as "linear independence" same as "...
2
votes
1answer
92 views

Sesquilinear Forms: Cauchy-Schwarz

This thread is related: Parallelogram Given a Hilbert space $\mathcal{H}$. Consider a quadratic form: $$q:\mathcal{H}\to\mathbb{C}:\quad q[\lambda\varphi]=|\lambda|^2q[\varphi]$$ Suppose it ...
4
votes
2answers
95 views

For which infinite dimensional real normed linear spaces $X$ , can we say that every infinite dimensional subspace of it is closed in $X$ ?

For which infinite dimensional real normed linear spaces $X$ , can we say that every infinite dimensional subspace of it is closed in $X$ ? Or , does every infinite dimensional normed linear space has ...
0
votes
2answers
195 views

Let V be a vector space of dimension n. Prove that no set of n - 1 vectors can span V.

I'm not sure I understand the question. As far as I understand it when it says vector space of dimension n, it signifies that there will be n amount of vectors; right? So basically it wants you to ...