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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1answer
40 views

How to show not closed under addition/closed under addition? (Specific exam question)

I need help in part (i) and the part where I am to show that $V_3$ is a linear space. I understand that I am required to show that, in part (i), $V_1$ ∪ $V_2$ is not closed under addition; I would ...
0
votes
0answers
34 views

Finding the basis and dimension of a subspace of the vector space of 2 by 2 matrices

I am trying to find the dimension and basis for the subspace spanned by: $$ \begin{bmatrix} 1&-5\\ -4&2 \end{bmatrix}, \begin{bmatrix} 1&1\\ -1&5 \end{bmatrix}, \begin{bmatrix} 2&-...
0
votes
0answers
22 views

Uniqueness theorem regarding Helmholtz decomposition of a Vector field

Helmholtz theorem wiki link states that given a smooth vector field $\pmb{H(x,y,z)}$, there are a scalar field $\phi (x,y,z)$ and a vector field $\pmb{G(x,y,z)}$ such that $$\pmb{H}=\pmb{\nabla} \phi +...
0
votes
1answer
43 views

Which vectors are obtainable by my function?

Imagine a disc with $N$ radially displaceable masses $m_g$. A total imbalance with respect to the center of the disc can be calculated as follows (using the respective radiuses $r_1,...,r_N$): $$\...
0
votes
1answer
23 views

Intersections of a Line & a Plane in 3d-Space [closed]

Given the line [x,y,z]:(1,-3,2)+t [-2,4,7] find planes to satisfy the following conditions: A plane which is intersected by the line at the point (3, -7, -5) I am unsure what to do with this, if I ...
2
votes
1answer
25 views

Commensurability for vector spaces

Let me start by saying I am a not a mathematician and I have not studied group theory (just a few brushes here and there) but after reading I have a very basic understanding of commensurability as ...
2
votes
3answers
74 views

Motivation for the dot product

We can motivate the cross product by considering a 3D vector perpendicular to two others. This results in 3 equations in 2 unknowns, i.e. a line of solutions, and... $\lambda(u_2 v_3 - v_2 u_3, ...
1
vote
1answer
41 views

Universal property of generating set for vector space

Let $V$ be a vector space over $F$, and $S$ a non-empty subset of $V$. We say that $S$ generates $V$ if every $v\in V$ can be written as finite $F$-linear combination of elements of $S$. I want to ...
1
vote
2answers
60 views

What does $V^*$ means?

What does it mean to have an "$f \in V^*$" in terms of a transformation? The chapter in the book it is in is about dimensions in vector spaces.
5
votes
2answers
74 views

Rational numbers as vectors in infinite dimensional space with the basis $( \log 2,\log 3, \log 5, \log 7, \dots, \log p, \dots) $

Since every natural number can be represented as $a=2^{n_1}3^{n_2}5^{n_3}7^{n_4}\cdots p_k^{n_k}\cdots$ it makes sense to represent natural numbers by vectors, using the properties of logarithms: $$\...
0
votes
0answers
30 views

Proving that if T∈Hom[V,W] has null space X∩Y, then T[X+Y]=T[X]⊕T[Y]

Here's my progress: If X and Y are subspaces of a finite-dimensional vector-space, then d(X+Y)+d(X∩Y)=d(X)+d(Y) , where d(A) is the dimension of A. Then, d(X∩Y)=d(X)+d(Y)-d(X+Y). But X∩Y is ...
1
vote
2answers
21 views

Quotient space example:

following this definition of wikipedia, I was trying to prove that this example is correct: Let $V = \left\{ (x_1,x_2,x_3) \right\}_{x_i \in \mathbb{R}}$ and $W = \left\{ (x_1,0,0) \right\}_{x_1 \in ...
0
votes
1answer
30 views

Smallest Eigen value of Matrix + other matrix less than greatest value eigen value of Matrix

I've been working on this problem for a little bit and I'm not sure if it can be proven with the given information. Any help would be greatly appreciated to either confirm or deny my suspicion. ...
2
votes
1answer
44 views

Why Gaussian elimination on the columns changes the column space?

This page on theorem 8.2 states that, Neither of the operations of the gaussian elimination changes the row space of an $m \times n$ matrix after applying the operation. It says later that this is ...
0
votes
0answers
7 views

Uniqueness of reflection stabilizing a finite generating set [duplicate]

Let $V$ be a finite dimensional real vector space, $v\in V$ . Define a reflection relative to $v$ to be a linear transformation which sends $v$ to $-v$ and fixes pointwise a subspace of ...
0
votes
0answers
27 views

linear decompositions of tensor product vectors

I have a vector space $V^{(1)} \otimes V^{(2)}$, a set of vectors $\{s_i\} \in V^{(1)}$ and $\{s_j\} \in V^{(2)}$, that span the respective spaces such that any one vector, e.g. $s_{i'}$, can be ...
-1
votes
1answer
31 views

Calculating the eigenvalues [closed]

I'm trying to understand the dynamics of the eigenvectors and the eigenvalues. My question is about formula for finding the eigenvalues. At 4:15(the athor starts the calculating at 1:30) of the given ...
-1
votes
1answer
19 views

dimension of span of vectors

It is obvious that the first two vectors are linearly dependent on one another, therefore one of them does not contribute to the dimension of the subspace. The null vector also clearly does not ...
2
votes
1answer
45 views

Decomposition and invariant subspaces

Let $\sigma$ be a linear operator on $V$. And $w$ is eigenvector with eigenvalue $1$, $W=\langle w\rangle$ (i.e. $\sigma|_{W}=\operatorname{Id}$), and also $\sigma|_{V/W}=\operatorname{Id}$. It ...
3
votes
0answers
34 views

Series and Linear Algebra

I have been studying series and I noticed that the Taylor series converges to a function using a polynomial basis and the Fourier series converges to a function in a given interval using a ...
0
votes
1answer
38 views

Projections satisfying $\| Px-Qx \| <\|x \|$ for nonzero $x$?

Let $V$ be a f.d inner product space with subspace $M,N$ and corresponding orthogonal projections $Q,P$. I need to prove that if $\| Px-Qx \| <\|x \|$ for all nonzero $x$, then $\dim M=\dim N$. As ...
0
votes
2answers
44 views

Proof that C is isomorphic to $End_C(V)$ with V the C-vector space C [closed]

I've read that $\mathbb C$ is isomorphic to $\mathrm{End}_{\mathbb C}(V)$, where $V$ is the $\mathbb C$-vector space $\mathbb C$. Does somebody know a proof for that? Greetings, Peter123.
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votes
0answers
33 views

Ways to determine a basis for $V$ given a basis for $V^*$ and the spaces $V$ and $V^*$

General Problem I'm looking for some ways to arrive at a basis for the vector space $V$ over a field $F$ given a basis for $V^*$ and a general vector in $V$. One approach I know of is to use that ...
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votes
3answers
27 views

Linear Algebra fining specific example coefficients [closed]

It can be shown that the vectors \begin{align} p_1(x)&=2x^2+4 \\ p_2(x)&=2x^2+x \\ p_3(x)&=−5x^2+6−4x \\ \end{align} are not linearly independent in $P_3$. Demonstrate this by finding a ...
2
votes
1answer
37 views

Can $f(x,z) = x^Tx + \sum\limits_{i = 1}^n \dfrac{x_ix_i}{z_i}$ be written with multiple inner products at the same time?

I am running into a very interesting phenomenon that I do not quite understand (Illustration of an example of so called subset of $\mathbb{R}^n$) For example, suppose we have a subset of $X \...
1
vote
1answer
24 views

How to find the base of union of 2 subspaces

Suppose we have 2 subspaces V,W. What is the base and the dimention of: V U W ? Its clear to me that V U W is not always a subspace! I was thinking about taking the base of U and the base of W and ...
0
votes
0answers
53 views

Convert velocity vector from body frame to world frame

I need to convert a 3D speed vector from a body frame to a world frame (ECEF). The velocity 3D vector I have is a linear body velocity, and I have the orientation of my object in radians in the world ...
2
votes
1answer
62 views

What is the point of “seeing” a set of polynomials or functions as a vector space?

I just had a course in linear algebra. It seemed that the main purpose is to lay the foundations of vector spaces, show ways of solving systems of linear equations and in the end, classify some ...
3
votes
1answer
45 views

On the difference between $\textbf{R}^{\{1,2,…,n\}}$, $\textbf{R}^{\{1,2,…,n+1\}}$, $\textbf{R}^{[0, 1]}$, and $\textbf{R}^\infty$

I'm working my way through Axler's "Linear Algebra Done Right" (3rd ed.), and I'm getting stuck on section 1.23, which says: If $S$ is a set, then $\textbf{F}^S$ denotes the set of functions ...
1
vote
1answer
34 views

Summation functional on a Hamel basis

Let $X$ be an infinite-dimensional Banach space. Is it possible to choose a Hamel basis $B$ of $X$ such that the linear functional defined by $f(b)=1$ ($b\in B$) was continuous?
2
votes
1answer
42 views

Vector spaces as bimodules

The usual definition of a vector space $V$ over $K$ is as an abelian group, on which $(K\setminus\{0\},\cdot)$ acts on the left, such that furthermore the operation of $K$ on $V$ is compatible with ...
0
votes
1answer
21 views

The dimension of $:W_1\cap W_2$

$$W_1=\operatorname{span}\left(\begin{pmatrix}1&1\\ 0&0\end{pmatrix},\begin{pmatrix}3&1\\ -1&0\end{pmatrix}\right)$$ $$W_2=\operatorname{span}\left(\begin{pmatrix}1&1\\ 1&0\...
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votes
0answers
10 views

Equality of vector & Directions

Is the direction 120° west of the South Axis same as the direction 30° north of the West Axis (both from a fixed observational point).
0
votes
2answers
43 views

Find $\lambda $ so the dimension of the vector subspace is 2.

$\left\{a,b,c\right\}\in \mathbb{R}^3$ are linearly independent vectors. Find the value of $\lambda $, so the dimension of the subspace generated by the vectors: $2a-3b,\:\:\left(\lambda -1\right)b-...
1
vote
1answer
52 views

Show that $\widetilde{f}$ is a nondegenerate map?

Noted by $V, W$ two vectors spaces over the same field $K$ of finite dimensions. Let $f:V\times V\rightarrow W$ a degenerate map. I would show that $$\widetilde{f}:\widetilde{V}\times \widetilde{V}\...
0
votes
1answer
19 views

difference between linear map basis and vector basis

A linear map can be represented as a matrix in a certain basis P. Similarly, given a vector space over a field, its basis can be found, say Q. How is the concept of P related to that of Q? Are they ...
1
vote
2answers
54 views

Whats the relation between the set of vector in a vector space to the field that this vector space is over?

We have a set of "vectors" (elements) in the set $V$ such that $(V,+,\circ)$ is said to be a vector space over some field $\mathbb{F}$. Let $F$ be the set of elements that consist the field $\mathbb{F}...
2
votes
1answer
45 views

A book on Vector Calculus with emphasis on geometrical intuition

I am a physicist trying to learn vector calculus in a way that is a mixture of the way mathematicians learn it with the way that physicist learn it in order to be able to learn Differential Geometry ...
0
votes
0answers
31 views

If I have n different eigenvalues prove their eigenvectors are linealy independent [duplicate]

Prove via induction that if $V$ is a vector space of finite dimension and T: $V\to V$ a linear operator with n different eigenvalues then the eigenvectors associated with them are linearly independent....
0
votes
1answer
48 views

In the context of linear algebra, is it possible for a vector space or a subspace to have a finite number of elements? [duplicate]

A vector space must satisfy closure under addition and multiplication. Sorry if this is obvious but does that mean that, assuming the normal rules of arithmetic and excluding the trivial examples like ...
4
votes
1answer
84 views

Converse of Schur's Lemma in finite dimensional vector spaces

I am trying to prove (or disprove) the converse of Schur's Lemma in finite dimensional vector spaces. I am not sure if it holds in this case, but I have tried to apply the idea that proves it in ...
1
vote
0answers
34 views

Why does power iteration generate almost dependent vectors?

On the Wiki page for Krylov subspaces: https://en.wikipedia.org/wiki/Krylov_subspace it states given a matrix $A$ and vector $b$, that the vectors $b, Ab, A^2b, A^3b, ...$ "soon become almost linearly ...
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votes
2answers
41 views

Example of a degenerate bilinear map?

I seek an example of a nonzero $\Bbb{R}$-bilinear map $f:V\times V\rightarrow W$ on a vector space $V$ (s.t: $\dim V<\infty$, $\dim W<\infty$) such that it is degenerate map, where $V$ and $W$ ...
0
votes
4answers
37 views

whether set of 3D vectors span space {(x, y, z) | x + y + z = 0}

Consider the set of (column) vectors defined by $X = \{x \in R^{3} | x_{1} + x_{2} + x_{3} = 0\}$, where $X^{T} = [x_{1}, x_{2}, x_{3}]^{T}$ , I need to prove whether(or not) given vectors, $[1, -1, 0]...
0
votes
1answer
60 views

Extension of mapping of subset to homomorphism

Is the following proposition true? Let $V$, $W\neq0$ be vector spaces over some field $F$ and let $S_v \subset V$. Then if every mapping $f:S_v \to W$ can be uniquely extended to homomorphism $g:V\to ...
0
votes
2answers
51 views

Meaning of Vector Space over $\mathbb{R}$ being a Subspace of $\mathbb{R^R}$

$\mathscr{P(\mathbb{R})}$ is the set of all polynomials with coefficients in $\mathbb{R}$. How are below sentences related and why? (1) $\mathscr{P(\mathbb{R})}$ is a vector space over $\mathbb{R}...
5
votes
4answers
175 views

Can someone explain this: “the set of subspaces of a vector space ordered by inclusion”

This is a claim on Wikipedia https://en.wikipedia.org/wiki/Partially_ordered_set I am not sure how to make sense of the claim What does it mean by ordered by inclusion? Inclusion as in $\subseteq$? ...
1
vote
0answers
39 views

Vector space complement to a multiplicatively closed subspace is an ideal

Let $V$ be a vector space over $\mathbb{C}$ of any dimension and suppose we have an associative multiplication $V \times V \to V$ making $V$ into a commutative ring with unity. Let $V=U \oplus W$ be a ...
0
votes
1answer
33 views

The map $f$ is degenerate or non-degenerate?

Let denote by $M_{3,2}(\mathbb C) $ the space of all $(3\times2)$-matrix of complex-dimension equal $6$ with basis $(E_{1},E_{2},E_{3},E_{4},E_{5},E_{6})$. Let $f$ a $\mathbb R$-bilinear skew-...
0
votes
2answers
39 views

Solving vector equation 3

Solve for $\bar{x}$ and $\bar{y}$ $$\bar{x}+\bar{y}=\bar{a},~~ \bar{x}\times \bar{y}=\bar{b},~~ \bar{x}.\bar{a}=1$$ Attempt: $\bar{x}+\bar{y}=\bar{a}$ dot by $\bar{a}$, we get $1+\bar{a}.\bar{y}=|...