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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22 views

How to Find the Dual Basis?

Suppose that $f_1 = 1, f_2 = x+3, f_3 = (x+3)^2, f_4 = (x+3)^3$ How would I approach finding a dual basis for this? I have no idea where to start!
4
votes
1answer
22 views

Prove that there is a base of $\mathbb R^4$ made of eigenvectors of matrix $A$

Matrix of linear operator $\mathcal A$:$\mathbb R^4$ $\rightarrow$ $\mathbb R^4$ is $$A= \begin{bmatrix} 1 & 1 & 1 & 1 \\ 1 & 1 & -1 & -1\\ 1 ...
0
votes
1answer
19 views

Confusion with some theorems leading to the canonical decomposition of an operator

Let $\Bbb K$ be a field and $R = \Bbb K[X]$ be the ring of polynomials with coefficients in $\Bbb K$. Let $\cal L_{\Bbb K}$$(V)$ denote, for a finite dimensional $\Bbb K$-vector space $V$, the set of ...
0
votes
2answers
29 views

What does the sum of subsets of a vector space mean?

On page $57$ of Second edition of Hoffman Kunze, the authors write Definition If $S_1, S_2, \dots, S_k$ are subsets of a vector space $V$, the set of all sums $$\alpha_1 + \alpha_2 + \dots+ ...
0
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0answers
19 views

How can I prove algebraic and topological dual spaces do not coincide in infinite dimensional normed vector spaces?

I've heard it's enough to give an example of a non continuous linear functional, but I'm kinda confused, because some definitions ask for "bounded" at infinite spaces, does bounded mean continuous in ...
0
votes
2answers
32 views

Prove/Disprove question on matrix vector multiplication and linear independence [on hold]

If $\left\{Bv_1, \ldots , Bv_k\right\}$ is a linearly independent set in $\mathbb{R}^k$ where $B$ is a $k \times n$ matrix, then $\left\{v_1, \ldots ,v_k\right\}$ is a linearly independent set in ...
0
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0answers
12 views

Show $\{\beta (v,w) =0 \Leftrightarrow \beta(w,v)= 0\} \:\:\:\: \Rightarrow \:\:\:\: \{\beta(w,u)\beta(u,v) = \beta (v,u) \beta (u,w) \}$

$\beta$ is a bilinear form on a $K$-vectorspace $V$. Now i have to show the following: if $\forall$ $v,w \in V$ $$\beta (v,w) =0 \Leftrightarrow \beta(w,v)= 0 $$ then $\forall u,v,w \in V$ ...
0
votes
0answers
30 views

Is there a term for the dimension of the annihilator of an element of an algebra?

Let $\mathcal{A}$ be a finite dimensional $R$-algebra, and for $x \in \mathcal{A}$ consider $\mathrm{Ann}(x) = \{ c \in \mathcal{A} \mid cx = 0 \}$. Is there a term for the dimension of this subspace? ...
1
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1answer
35 views

Diference between dual spaces

What is the diference between Algebraic Dual Space and Topologic Dual Space in Normed Vector spaces with $dim=\infty$
2
votes
1answer
55 views

Norms on $\mathcal{P}_N$ Vector Space of Polynomials up to Order N

$\|p\|_\infty :=\sup_{x\in [0,1]}|p(x)|$ and $\|p\|_{L^1}:=\int_0^1 |p(x)| dx$. As the space of real-valued polynomials on $[0,1]$ up to order $N$ is a $N+1$ dimensional vector space and ...
1
vote
2answers
23 views

Find subspace $T$ of space $\mathbb R^3$ so that $\mathbb R^3=S \oplus T$

I have one problem. I am sure it is not complicated, but I only need help to see am I, at least, on the right path. Problem: Let $S=Span\{(0,-2,3),(1,1,1),(2, -2, 8)\}\subseteq \mathbb R^3$. Find ...
1
vote
1answer
25 views

Sylvester's argument for bilinear functions

Let $V$ be a vector space of dimension $n$ and let $b:\colon V \times V\to \mathbb{R}$ be a symmetric bilinear function. Sylvester's theorem says that there exists a basis of $V$ with respect to ...
2
votes
1answer
22 views

Basis for the space of linear transformations $L(\Bbb R^3,\Bbb R_3[x])$

How do I build a basis for the vector space $L(\Bbb R^3,\Bbb R_3[x])$? This is the vector space of all linear transformations that goes from $\Bbb R^3$ to the space of polynomials of degree 3 ...
3
votes
2answers
34 views

Wedge product is nondegenerate symmetric bilinear form

Let$$f: \Lambda^k(\mathbb{R}^n) \times \Lambda^{n - k}(\mathbb{R}^n) \to \mathbb{R}, \quad f(\alpha, \beta) = \alpha \wedge \beta.$$How do I see that $f$ is a nondegenerate symmetric bilinear form?
1
vote
2answers
44 views

Prove that $U$ is a vector-subspace

If $U$ is the set of all matrices that are commutative with the matrix $A$, show that $U$ is a vector subspace of the space $M^\mathbb{R}_{3\times 3}$ $$A=\begin{pmatrix}2&0&1\\ ...
0
votes
1answer
18 views

Problem with change of basis of an polynomial.

Good morning, i have a problem solving this: Express $a_{0}+a_{1}x+a_{2}x^{2}$ in terms of basis: $1,x-1,x^{2}-1$ I make this: ...
1
vote
2answers
41 views

Does there exists an additive group homomorphism between two $K$-vector space that is not $K$-linear

My question is: Give me a field $K$. Can we always find two $K$-vector space $V_{1}$, $V_{2}$ and a map $f:V_{1}\rightarrow V_{2}$ such that: (1) If we view $V_{1}$, $V_{2}$ as additive group, then ...
2
votes
2answers
102 views

What does a norm of a polynomial space mean?

When talking about polynomial vector space, the following example was provided. A polynomial of degree $n$ in two variables is $$p(X)=\sum_{0\leq k+j \leq n} a_{j,k}x_1^jx_2^k$$ where $k+j=n$ and ...
1
vote
2answers
29 views

Prove that a union of bases for $S$ and $T$ is basis for $S + T$

Let $S$ and $T$ be subspaces of a vector space $V$. Assume that $B = \{ b_i | i \in I \}$ is a basis for $S \cap T$. Now, extend $B$ to a basis $A \cup B$ for $S$ where $A = \{ a_j | j \in J \}$ and ...
0
votes
0answers
19 views

Same vector space for arbitrary independent vectors?

If we use n linearly independent vectors x1,x2...xn to form a vector space V and use another set of n linearly independent vectors y1,y2...yn to form a vector space S, is it necessary that V and S are ...
0
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0answers
11 views

What are the Geometric Properties of Non Integer Vector Spaces?

I found a paper from Princeton called "Axiomatic Basis for Spaces with Non Integer Dimension" that presents five axioms and then starts to create a framework similar to what I'd think the subject ...
0
votes
0answers
33 views

The precise definition of Cartesian coordinate and Euclidean space?

I'd searched them for a while, but still have not found a clear and unity definition on it. The problem really confused me. What is the precise definition of Cartesian coordinate and Euclidean space? ...
4
votes
3answers
140 views

Doubt with vectorial spaces (Basis and dimension)

Good night, i'm working in a problem, i need an basis and the dimension of the space. $a_{1}=(1,0,0,-1),\:a_{2}=(2,1,1,0),\:a_{3}=(1,1,1,1),\:a_{4}=(1,2,3,4),\:a_{5}=(0,1,2,3)$ I make this: $\left[ ...
0
votes
2answers
30 views

Prove that a linear mapping between vector spaces is an open mapping iff

Let $(N,|| \ ||)$ and $(N_1,||\ ||_1)$ be normed vector spaces and $f$ a linear mapping of $N$ into $N_1$. Prove that $f$ is an open mapping if and only if $\forall$ $n \in \Bbb N $, $B_r(0) ...
3
votes
0answers
27 views

Sum of projection operators

Given $p_1, ..., p_n$ $n$ projection operators on the vector space $E$ such that $\sum_{i=1}^n p_i$ is a projection operator. How to show that $\forall i,j \text{ s.t. } i \neq j, p_i \circ p_j = 0$ ...
0
votes
0answers
20 views

mean-deviation form, why orthogonal?

This is from my textbook Why the column of the new design matrix are orthogonal? for example, let say $A=\begin{pmatrix} 1& 1& 4\\ 1& 2& 0\\ 1& 3& 2 \end{pmatrix}$ ...
1
vote
2answers
17 views

How can one characterise the number of linear combinations of m > 2 linearly independent vectors that map onto the same point in the plane?

I have m > 2 vectors v in the plane, any two of which are linearly independent to each other. Any two of these vectors are enough to fill the plane. My question is this: How can one characterise ...
1
vote
0answers
25 views

Notation: rotation matrix with a condition

I'm building a space simulation & am using this resource for converting Keplerian Orbit Elements to Cartesian Co-ordinates. The notation for step 6 has me slightly confused: Is the top part ...
0
votes
1answer
26 views

Vector norm lemma and proof

I have a question from Numerical linear algebra book by Trefethen & Bau : Let $\|\cdot\|$ denote any norm on $C^m$. The corresponding dual norm $\|\cdot\|'$ is defined by the formula ...
0
votes
0answers
15 views

Finding altitude and azimuth with an accelerometer and magnetometer

I posted this in the astronomy stack exchange forum, but considering that it is a very math intensive question I figured there could also be people on here that could help. For a project with my ...
0
votes
1answer
24 views

finding inner product

This is from my textbook: I don't know how to tell whether the spanning set are actually orthogonal. The textbook's solution is like this, forexample, to see if $P_0(t)$ and $P_1(t)$ are orthognal, ...
3
votes
1answer
30 views

Uniqueness of endpoints of half-open line segments in linear spaces.

I try to solve the following exercise, which is Exercise 1.18 in Robert Megginson's An Introduction to Banach Space Theory. Let $X$ be a linear space, and define for any $x_1, x_2 \in X$ the line ...
0
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0answers
26 views

Prove that then the linear mapping is surjective [closed]

Let in the field $F$: $2 \ne 0$. Let $Q$ is a nondegenerate quadratic form on a finite-dimensional vector space $V$. Suppose that $Q(v) = 0$ for some nonzero vector $v \in V$. Prove that then the ...
0
votes
2answers
25 views

Find a point 90° left or right from a point (x,y,z) in a 3D space.

How can I find a point which is 90° left or right from a point (x,y,z) in a 3D space? for example if I have the point $(x,y,z)$ how to find $(x1,y1,z1)$ and $(x2,y2,z2)$.
0
votes
1answer
23 views

vector space homomorphism for $Map(\mathbb{F}_{5} , \mathbb{F}_{5})$

I'm currently stuck at a mathematical problem and I really don't know where to start.. Since I'm not an expert in Algebra over finite fields... It goes "Define a $\mathbb{F}_{5}$-vector space ...
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votes
0answers
7 views

How can I find the necessary speed and speed of rotation for a problem from a parametric equation?

I have been given the following questions for a project that I am currently working on: Questions 1 to 8 I have completed questions 1 through 6 but have no idea how to do questions 6 or 7 after ...
0
votes
1answer
24 views

Intersection of normed speces and desity

Let $(X_n, \|\cdot\|_n)$ be a sequence of normed spaces. My first question is, whether it is possible to norm $X= \cap_n X_n$. My idea would be to take $\|\cdot\|_X = \sup \|\cdot\|_n$ if it is ...
0
votes
1answer
25 views

Finding span of intersection of two vector subspaces

I was trying to follow this answer, but as the comment to that answer suggests, there's a problem with dimensions, and that's exactly where I'm stuck. More concretely, I have subspaces $U$ and $W$, ...
1
vote
2answers
21 views

Does anti-Hermtian matrices from a vector space?

My book states that $n\times n$ anti-Hermitian matrices $T^\dagger = -T$ form a real vector space. But the identity matrix is not anti-Hermitian and hence doesn't belong to this set. Is my book wrong? ...
1
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0answers
16 views

Solution space of semilinear equation

I found the following lemma and the corollary in a paper and I don't know how to prove them. Therefore I was wondering if one of you could help me. Let $E$ be a field, $ \sigma: E \rightarrow E$ ...
1
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0answers
11 views

Change of Coordinates and Basis

Let $P_{2}(\mathbb{R})$ denote the vector space of real polynomial functions of degree less than or equal to two and let $\beta := \{p_{0}, p_{1}, p_{2}\}$ denote the natural basis of ...
2
votes
1answer
15 views

If $M$ is a simple $R$-module, and an $F$-space, why does $End_F(M)\cong M^{\oplus\dim_F(M)}$?

Suppose a ring $R$ is an $F$-algebra for $F$ a field, and $M$ is a simple $R$-module and a finite dimensional $F$-vector space. We can endow $\operatorname{End}_F(M)$ with an $R$-module structure by ...
1
vote
2answers
21 views

Understanding a certain step in a proof about a basis of a vector space

This is a theorem from Roman's textbook "Advanced Linear Algebra"(p.$48$). Theorem $1.9.$ Let $V$ be a nonzero vector space. Let $I$ be a linearly independent subset of $V$ and let $S$ be a ...
0
votes
0answers
39 views

Algebra quotient space homomorphism

I have to prove the following; Let $A$ be an algebra over a field $K$. If $I \subset A$ is an ideal, then there exists a unique algebra structure on the quotient vector space $A/I$ such that the ...
0
votes
0answers
27 views

Linear algebra textbook for quantum computing?

I'm looking for an recommendation for a linear algebra textbook specifically to give me the background for learning about quantum computing, and quantum mechanics more generally. In particular, none ...
0
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2answers
22 views

If every non-zero vectors be the eigenvector of a real matrix $A$, prove that $A$ is the scalar matrix $\lambda I_n$.

If every non-zero vectors in $\mathbb{R}^n$ be the eigenvector of a real $n \times n$ matrix $A$ corresponding to a real eigenvalue $\lambda$, prove that $A$ is the scalar matrix $\lambda I_n$. I ...
0
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0answers
43 views

The set of all $n\times n$ matrices A such that the $A^T = A^{-1}$ is a subspace of the vertor space $M_n(\mathbb{R})$

I think the set of $n \times n$ matrices such that $A^T = A^{-1}$ is not a vector space since it doesn't have $0$. How do I show that it's not a subspace?
2
votes
1answer
43 views

Is the converse of the Pythagorean Theorem false for complex inner products?

I was thinking about the converse of the Pythagorean theorem: $\lVert x + y\rVert^2 = \lVert x\rVert^2 + \lVert y\rVert^2 \implies x \perp y$ Does this hold if the inner product $\langle ...
0
votes
0answers
25 views

Simple excercise on linear transformations - confused

A Linear tranformation L in $\mathbb R^3$ with matrix $$ L_b^b = \left(\begin{matrix} 1 & 0 & 5 \\ 0 & -2 & 2 \\ 1 & -2 & 7 \end{matrix}\right)$$ and basis $b = \{ (1,0,2), ...
1
vote
1answer
48 views

Dimension of subspace of $\text{End}(\mathbb{R}^5)$

I'm doing a problem which presented me with a basis for some $U\subseteq\mathbb{R}^5$ where $\dim U=3$ (I can give it explicitly if that helps but I do not think it matters). The question is this: ...