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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3answers
161 views

basis functions do not lie in the space they form

For example, any continuous function in $\mathbb{L}^2(-\infty,\infty)$ space can be expanded by delta functions $\delta(x-a)$ or Fourier basis $e^{ikx}$. However, the basis functions, both ...
4
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1answer
348 views

Center of Clifford Algebra depending on the parity of $\dim V$?

While reading about the structure of Clifford algebra, there were two facts listed as bullet points about the center of Clifford algebra based on the parity of the dimension of the underlying vector ...
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1answer
86 views

To prove that the dimension of $V$ is $d_1^2 + \ldots + d_k^2$

Let $A$ be an $n \times n$ diagonal matrix with characteristic polynomial $$(x - c_1)^{d_1} \cdots (x - c_k)^{d_k} , $$ where $c_1,\ldots,c_k$ are distinct. Let $V$ be the space of $n \times n$ ...
4
votes
1answer
324 views

Splicing together Short Exact Sequences

Given an exact sequence of vector spaces $$\cdots\longrightarrow V_{i-1}\longrightarrow V_{i}\longrightarrow V_{i+1}\longrightarrow\cdots$$ I want to show that it is the same as having a collection of ...
4
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1answer
148 views

What does a subspace spanned by another subspace and a vector mean?

What does a subspace say A spanned by another subspace B and a vector x mean ? Does that imply anything about a basis or does it just mean that every vector in subspace A is either present in ...
4
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1answer
154 views

Is following subset W of V also a subspace?

Under following conditions $ a, b \in \mathbb{R}, V = \mathbb{R}^{2}, W = \{(x, y)\ |\ ax + by = 0 \} $ is W a subspace of V? I know the basics, but how would I prove that addition and ...
4
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1answer
121 views

Proof of $X\cup Y\neq V$

Suppose $X,Y$ are subspaces of dimension $n-k$ of the vector space $V$ of dimension $n$. Why is it always true that $X\cup Y\neq V$?  I can show this by arguing that if $X=Y$ then clearly by the ...
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2answers
226 views

Dot product of two vectors

How does one show that the dot product of two vectors is A · B = |A| * |B| * cos(Θ) ?
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4answers
446 views

Let $V$ be a $k$-vector space of dimension $n$ and $T: V \to V$ a linear map of rank 1. Show that either $T^2 = 0$ or that $T$ is diagonalisable?

I am not good with vector spaces so I would be grateful for any help. As I've been told I need to take $v \in \mathrm{Im}(T)$, $v\neq 0$, and show that if $T(v) = \mathbf{0}$ then $T^2 = 0$. But if ...
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1answer
42 views

Dual space and linear functional

The problem is this: Let $V$ be a vector space over $\mathbb{K}$ and $v \in V$. Show that if $f(v) = 0, \forall$ $f \in L(V, \mathbb{K})$, then $v=0$. It's a problem of a book I'm using to study Dual ...
4
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1answer
116 views

What does the symbol: $\mathcal{F}(S,F)$ in linear algebra mean?

I have a problem in linear algebra course, and I'm looking to solve it by myself, but I'm confused with notation since my teacher never mention it in class. It says: Let $S$ be a nonempty set and ...
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2answers
2k views

Vector space of polynomials

Do all polynomials $ax^3 + bx^2 + cx + d$ with a root at $x=1$ form a vector space? Do the coefficients $(a,b,c,d)$ form a vector space? My reasoning: Since $x=1$ is a root, we can't have $(a,b,c,d)$ ...
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1answer
2k views

Relation between cross-product and outer product

If inner products ($V$) are generalisations of dot products ($ \mathbb{R}^n$), then are outer products ($V$) also related to cross-products ($ \mathbb{R}^3$) in some way? A quick search reveals that ...
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1answer
525 views

direct products and direct sums for matrices and for vector spaces

I was wondering what relations and similarities are between direct product for matrices and direct product for vector spaces? Or do they just unfortunately and somehow misleadingly happen to have the ...
4
votes
2answers
63 views

Is $S=\{(1,t)\mid t\in \mathbb{R}\}$ a subspace of $\mathbb{R}^2$?

My professor introduced subspaces of $\mathbb{R}^n$ today and I don't think I understand them very well. He posed this question as an example: Is the set $S=\{(1,t)\mid t\in \mathbb{R}\}$ a ...
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2answers
59 views

Another linear algebra question

I have no idea how to start the following question. Any help will be greatly appreciated. (a) Let $A$ be a $n\times n$ matrix and let $a_1,...,a_n$ be the rows of $A.$ Suppose $y=(y_1, ..., y_n)$ is ...
4
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3answers
254 views

How do you construct the quaternion and the multiplication rules, like Hamilton did?

So, I understand complex number multiplication, and how it represents $2D$ rotations. What I don't understand is, how you add two more imaginary numbers $j$ and $k$, and get $3D$ rotations. I believe ...
4
votes
1answer
280 views

Image of unit ball dense under continuous map between banach spaces

I am assuming that the following problem will require the open mapping theorem, or maybe the closed graph theorem. Any help that can be given will be deeply appreciated. The statement is the ...
4
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2answers
108 views

Normed linear space and linear functional

Let $X$ be the normed linear spaceof sequences of reals that have only finitely many non-zero terms. Given $x = \{x_n\} \in X$, define $$f(x) = \displaystyle \sum_{n=1}^{\infty} x_n$$ I think that it ...
4
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1answer
210 views

Looking for proof that an open set in vector space contains the sum of two open sets.

Problem: To show that, in a topological vector space, for a given neighborhood of zero $W$, there exist two neighborhoods of zero, $V_1$, $V_2$, whose sum is contained in the first neighborhood, ...
4
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1answer
412 views

When are two diagonal matrices congruent?

This is probably a question that does not admit a simple answer. However, I'd like to know whether there exist criteria that determine when two diagonal matrices are congruent. I have the suspicion ...
4
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1answer
290 views

Direct sum $\Rightarrow$ Direct Integral, Tensor product $\Rightarrow$?

Is there a way to define a tensor product over a measure space(=index set) with a continuous measure for Hilbert spaces? For the sum we have the notion of a direct integral, here.
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1answer
52 views

Left adjoint to direct sum?

In the category of vector spaces, is there some endofunctor $F$ satisfying $$\mathrm{Hom}_k(M,\underset{i \in I}{\bigoplus} k) \cong \mathrm{Hom}_k(F(M),k)$$ for every $k$-vector space $M$?
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1answer
47 views

Projection onto space in $L^2([0,1])$ gives shorter length

Let $f_1,f_2,\ldots,f_n\in L^2([0,1])$, and let $V$ denote their span. Let $P:L^2([0,1])\rightarrow V$ be the projection onto $V$. Let $g\in L^2([0,1])$. Suppose also that $g\in L^p([0,1])$ for some ...
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2answers
106 views

Show that $F+G$ is closed when $G$ a closed subspace of normed space $E$ and $F$ a finite dimensional subspace of $E$.

Question: Let $E$ be a normed space. Let $G$ be a closed subspace of $E$ and let $F$ be a finite dimensional subspace of $E$. Show that $F+G$ is a subspace of $E$ and is closed. I'm having trouble in ...
4
votes
3answers
128 views

Invertibility in a finite-dimensional inner product space

Let $T$ be an invertible linear operator on a finite-dimensional inner product space. I just want a hint as to how I should prove that $T^{\star}$ is also invertible and $( T^{-1} )^{\star} = ( ...
4
votes
1answer
131 views

$V=W_1\oplus\cdots\oplus W_k$ if and only if $\dim(V)=\sum{\dim(W_i)}$

If $W_1,\dots, W_k$ are subspaces of a finite dimensional vector space $V$ such that $W_1+\cdots+W_k=V$, and I want to show that $V=W_1\oplus\cdots\oplus W_k$ if and only if $\dim(V)=\sum{W_i}$, then ...
4
votes
1answer
76 views

Revisted: $T^{**}\circ \varphi_1 = \varphi_2\circ T$

So, I'm trying to show that $T^{**}\circ \varphi_1 = \varphi_2\circ T$ where $\varphi_1 : T\rightarrow V^{**}$ and $\varphi_2 : W\rightarrow W^{**}$. Also, $T^{**} : V^{**}\rightarrow W^{**}$, and yes ...
4
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3answers
132 views

$T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$

How should one prove that there exists a linear map $T:V\rightarrow W$ such that $N(T)=V'\subset V$ and $R(T)=W'\subset W$ if $\dim(V')+\dim(W')=\dim(V)$, where $V$ and $W$ are finite-dimensional ...
4
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1answer
193 views

Smallest/Minimal bases of a topological space

The smallest possible cardinality of a base is called the weight of the topological space. I was wondering if all minimal bases have the same cardinality, and if every base contains a subset whose ...
4
votes
1answer
136 views

Differentiation continuous iff domain is finite dimensional

Let $A\subset C([0,1])$ a closed linear subspace with respect to the usual supremum norm satisfying $A\subset C^1([0,1])$. Is $D\colon A\rightarrow C([0,1]), \ f\rightarrow f'$ continuous iff $A$ ...
4
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1answer
175 views

Is there a name for this $k$-fold vector product?

Let $V$ be a set of vectors of length $n$. Define a $k$-fold product on $V$, $$ \Upsilon(\{v_1,\ldots,v_k\}):=\sum_{j=1}^n\prod_{i=1}^k v_{ij}, $$ where $v_i\in V$ and $v_{ij}$ is the $j^\text{th}$ ...
4
votes
1answer
681 views

Cross product of vectors as a determinant: valid matrix operation?

"The definition of the cross product can also be represented by the determinant of a formal matrix." —Wikipedia This seems like a hack to me—something of much practical use but ...
4
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1answer
790 views

Finding normal vectors of polygons

I have the following diagram: I want to find the normal vector for the polygon of points $abc$ and the plane highlighted in red with the points $bcde$. To find the normal vector for the polygon of ...
4
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1answer
43 views

Duality mappings on finite-dimensional spaces

I have a few questions regarding some concepts in a book "nonlinear partial differential equations by Roubicek" I am studying. The following is from the text. "Let $V$ be a separable, reflexive ...
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votes
1answer
86 views

Euclidean space without orthonormal basis

I've been thinking about: Problem Give an example of a nonseperable Euclidean space which has no orthonormal basis. My Argument I know if a Euclidean space $R$ has at most countable basis, ...
4
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1answer
65 views

All Invariant Subspaces of a Linear Transformation

I got this problem: Let $T:\mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation such that all it's eigenvalues are 1, 2 and 3 and the corresponding eigenvectors are $v_1, v_2$ and $v_3$ ...
4
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1answer
118 views

A basic question on linear maps and upper triangular form

Let $S$ and $T$ be two linear maps from $V$ to $V$ ($V$ complex vector space) such that $ST=TS$. I need to prove that there exists a basis with respect to which both the matrices are in ...
4
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1answer
65 views

Is my linear algebra proof correct

I proved that for subspaces $U_i$ of $V$ the inequality $\mathrm{dim}(U_1 + ...+U_m) \le \mathrm{dim}(U_1) + ... + \mathrm{dim}(U_m)$ holds. I proved it as follows (would you please tell me if my ...
4
votes
2answers
53 views

is this subset a subspace - redux

OK, I have been bothering people here with this for days and with luck I finally have this. People have helped a lot here so far. (Doing these examples is I hope helping me learn the proofs, but I ...
4
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1answer
52 views

Given $T(A) = A^t$ in $M_{n\times n}(\mathbb R)$. Find the polynomials and find if it's diagonalizable

Given the vector space $M_{n\times n} (\mathbb R)$ and a transformation $T(A) = A^t$ (transpose): Find $m_T$, $P_T$ (the minimum polynomial and the characteristic polynomial respectively.) ...
4
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1answer
75 views

Inner product spaces that are isometrically isomorphic

I know this is a fundamental result in linear algebra, and although it is referenced in my textbook, it does not have a proof for it. I was wondering if someone could help me out: Let $V$ and $W$ be ...
4
votes
1answer
218 views

Tensor product of real numbers over the rationals

How do I show that $\mathbb{R}\otimes_{\mathbb{Q}}\mathbb{R}\not\cong\mathbb{R}$ as $\mathbb{R}$-vector spaces? Possible approaches I can think of (but can't implement) are to show that this tensor ...
4
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2answers
184 views

Representations of Direct Sum of Lie Algebras

I'm trying to prove the following. Let $\frak{g}$ and $\frak{h}$ be (semisimple) Lie algebras. Then every representation $d$ of $\frak{g}\oplus\frak{h}$ is the tensor product of representations $d^1$ ...
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2answers
378 views

Vector Space of Polynomial Functions

Suppose that $p_0, p_1, ..., p_m$ are polynomials in $P_m(F)$ such that $p_i(2)=0$ for each $i$. Prove that the set of vectors $p_0, ..., p_m$ is linearly dependent. Note that $P_m(F)$ denotes the ...
4
votes
2answers
134 views

Vectors transformation

Give a necessary and sufficient condition ("if and only if") for when three vectors $a, b, c, \in \mathbb{R^2}$ can be transformed to unit length vectors by a single affine transformation. This is ...
4
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1answer
47 views

number of 1-to-1 linear functions on vectorspaces over finite fields

This is not a homework. I just ask this question myself and thought it would be easy to figure out. But I did not get the solution. Let $\mathbb{F}$ be a finite field with $|\mathbb{F}|=q$. Consider ...
4
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1answer
120 views

Product of binomial coefficient as a basis

I am stuck with the following problem. Every polynomial of degree $d$ can be expressed as $$ p(x) = p_d \cdot \binom{x}{d}+ p_{d-1} \cdot \binom{x}{d-1} + \cdots + p_0 \binom{x}{0} $$ What is the ...
4
votes
1answer
158 views

“Algorithmic” proofs in linear algebra

Although I am new to linear algebra, I want to study it with as much rigor as possible. After searching around, I picked up Halmos' Finite Dimensional Vector Spaces and Axler's Linear Algebra Done ...
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votes
4answers
756 views

Sphere tangent to a plane

Find the equation for a sphere with center $(\alpha,\beta,\gamma)$ tangent to the plane $ax + by + cz = d$. The sphere is $(x-\alpha)^2 + (y-\beta)^2 +(z-\gamma)^2 = r^2$ and I understand that some ...