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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$N$-dimensional linear operator is normal, Lagrange interpolation?

Is there a way to see that an $N$-dimensional linear operator $A$ is normal if and only if $A^\dagger$ can be represented as a linear combination of $I, A, A^2, \dots, A^{N-1}$ using Lagrange ...
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1answer
38 views

Is any linear combination of arbitrary elements in a vector space also arbitrary?

Assuming $K$ is some vector space, is it valid to say the following: If $a, b, c \in K$ are arbitrary and $\gamma$ and $\phi$ are scalars, then $a+b$, $a+c$, $a+b+c$, $\gamma a$, $\gamma b$, ...
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3answers
64 views

Proving that $V$ is a vector space, if $a+b=ab$ and $a *b=a^b$.

I am currently studying Halmos' "Linear Algebra Problem Book" and am stuck on problem 21(4). Let $V$ be the set $\mathbb{R}_+$, and let $F$ be the set $\mathbb{R}$. Let's define the sum of two ...
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1answer
55 views

Subsets of a vector space [closed]

What is the definitions of subset and subspace of a vector space? Give some examples. What does the word "span" mean? As example, I have a $R^3$ as a vector space. Can I get a subset generated ...
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25 views

Finding transformation with respect to a basis

Let $T:R^3 \rightarrow R^3$ be a non-invertible linear transformation that's represented with respect to the base: $ B = ((1,0,1),(0,1,-1),(1,-1,0))$ By the matrix: $$[T]_B=\begin{pmatrix} 1 & 0 ...
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1answer
29 views

Set a new length for a vector?

I never encountered such action. Can someone explain this on page 47? The programmer uses a "SetLength" function on a 3-dimensional vector. Here's the statement: ...
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2answers
38 views

Example of straight line and a point in $\Bbb R^3$ such that there are infinitely many planes passing through it

Give an example of a straight line $l$ in $\mathbb R^3$, given by a system of two equations, and a point $(a,b,c)\in \mathbb R^3$ such that there are infinitely many planes in $\mathbb R^3$ passing ...
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3answers
62 views

Isomorphism of vector spaces

There is an example given in my lecture notes, which I feel a bit uncertain about $ \{ $functions$ \{1,2,...,n \} \to \mathbf F \} \cong \mathbf F^n $ by the map $ f \mapsto (f(1),f(2),...f(n))$ ...
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2answers
28 views

Given two basis, find the transformation matrix from one to another

I have these two basis of $M^R_{2x2}$: $C= (\begin{pmatrix}1 & 0 \\ 0 & 0\end{pmatrix}, \begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}, \begin{pmatrix}0 & 0 \\ 1 & ...
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3answers
26 views

Showing Vector Space

Let $S = \{(x,y,z)\in R^3 : 2x-3y+5z=0\}.$ show that S is a real vector space using the standard operations on $R^3$ Having trouble showing S is closed under vector addition because of the ...
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2answers
26 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, ...
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3answers
67 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 ...
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3answers
81 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 ...
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2answers
46 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 ...
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3answers
61 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 ...
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1answer
20 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). ...
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1answer
40 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 ...
2
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3answers
34 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 $T$ ...
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1answer
24 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.
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2answers
37 views

Prove $\{a(x,y,z)=(ax,y,z)\}$ is a vector space [closed]

The set $R(R^3,+,\centerdot)$ with usual addition and scalar multiplication $a(x,y,z)=(ax,y,z)$ is a vector space? I don't get it, he gives a counter example proving that it doesn't hold for the ...
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5answers
196 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 ...
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2answers
220 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, ...
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0answers
28 views

Dimension of the set of all linear maps which preserve some subspaces

Let $V$ be a $10$-dimensional real vector space and $U_1$ and $U_2$ two linear subspaces such that $U_1 ⊆ U_2$, $\dim_\mathbb R U_1 = 3$ and $\dim_\mathbb R U_2 = 6$. Let E be the set of all linear ...
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0answers
50 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 ...
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1answer
52 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
131 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 ...
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2answers
75 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 ...
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2answers
42 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 ...
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3answers
28 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 $H$. ...
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1answer
30 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 $[T]_{\beta}$ ...
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3answers
48 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 ...
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0answers
54 views

Spinors of Linear Algebra

Let $V$ be an $n$ dimensional real vector space and $V^*$ be the dual vector space. We have a non degenerate inner product $(\centerdot,\centerdot)$ on $V\oplus V^*$ such that $(v+\xi , ...
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2answers
59 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 ...
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1answer
18 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 ...
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1answer
27 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
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1answer
35 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 ...
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2answers
36 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, ...
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2answers
36 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 , ...
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2answers
32 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
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1answer
26 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 ...
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0answers
84 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 ...
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1answer
32 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 ...
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0answers
23 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 ...
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0answers
14 views

Parameterization of a “concurrent line”

What is a valid parameterization for a general, real intersection of two surfaces: $$ f(x,y,z) = 0, \, g(x,y,z) =0 ? $$ For particular cases we eliminate a coordinate if possible and use the form ...
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3answers
61 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 ...
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3answers
53 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 ...
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1answer
86 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 ...
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3answers
32 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 ...
2
votes
4answers
61 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 ...
1
vote
4answers
99 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 ...