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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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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5answers
611 views

Canonical Isomorphism Between $\mathbf{V}$ and $(\mathbf{V}^*)^*$

For the finite-dimensional case, we have a canonical isomorphism between $\mathbf{V}$, a vector space with the usual addition and scalar multiplication, and $(\mathbf{V}^*)^*$, the "dual of the dual ...
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0answers
128 views

Why is the radical of a Clifford algebra generated by the kernel of the associated symmetric form? [duplicate]

I was recently reading through Jacobson's Basic Algebra. I got to the section on Clifford algebras, and have the following question. Let $Cl_\omega$ be the Clifford algebra with bilinear symmetric ...
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2answers
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Finite fields as vector spaces

I'm having great difficulty understanding this topic. Can someone concretely explain what it is meant by thinking of $GF(q^2)$ ($q$ a prime power) as a two-dimensional vector space over its subfield ...
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6answers
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Prove in full detail that the set is a vector space

So I'm doing a review test and I have this problem: Prove in full detail, with the standard operations in R2, that the set {(x,2x): x is a real number} is a ...
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2answers
173 views

Tangent space and tangent vectors

As I have heard, tangent vector to a smooth manifold $M$ in $p \in M$ is the operator $D_{\xi}$:$f \to D_{\xi}f$, where $f$ is a smooth function $f: M \to R$, with the following properties: ...
3
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1answer
58 views

Exact sequence arising from symplectic manifold

Let $M$ be a symplectic manifold, why is the following sequence exact? $$0\to \mathbb{R} \to C^\infty (M)\to A\to 0$$ Here $A$ is the set of global Hamiltonian vector fields.
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5answers
236 views

Dimensions: $\bigcap^{k}_{i=1}V_i \neq \{0\}$

Let $V$ be a vector space of dimension $n$ and let $V_1,V_2,\ldots,V_k \subset V$ be subspaces of $V$. Assume that \begin{eqnarray} \sum^{k}_{i=1} \dim(V_i) > n(k-1). \end{eqnarray} To show that ...
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1answer
393 views

A stronger statement of Riesz's lemma

Riesz's lemma state that If $Y$ is a proper, closed subspace of a normed space $X$, then for any $\epsilon>0$, there exists $x$ in the closed unit ball of $X$ such that $d(x,Y)>1-\epsilon$. ...
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votes
3answers
196 views

Possible proof for the relation involving matrix trace

Suppose a diagonal matrix $D\in\mathbb{R}^{n\times n}$ is given, with all its entries $d_{ii}\geq0$, for all $i$. Is it possible to prove ...
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3answers
87 views

Show that $T\to T^*$ is an isomorphism (Where T is a linear transform)

I think I solved it, but I used a dirty trick, I'd like someone to review it, that would be great. Let $X,Y$ be linear spaces over field $F$. and $T:X \to Y$ a linear transformation.For each $T$ we ...
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1answer
412 views

Proof about orthogonal subspaces

There is a vector space E , which is also finite-dimensional, and it contains subspaces V1 and V2. I need help proving that: 1. ( V1∩ V2)0 = V10 + V20 2. ( V1+ V2)0 = V10 ∩ V20 Thanks!
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3answers
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What is the norm measuring in function spaces

In spatial euclidean vector spaces norm is an intuitive concept: It measures the distance from the null vector and from other vectors. The generalization to function spaces is quite a mental leap (at ...
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6answers
970 views

Showing $1,e^{x}$ and $\sin{x}$ are linearly independent in $\mathcal{C}[0,1]$

How do i show that $f_{1}(x)=1$, $f_{2}(x)=e^{x}$ and $f_{3}(x)=\sin{x}$ are linearly independent, as elements of the vector space, of continuous functions $\mathcal{C}[0,1]$. So for showing these ...
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votes
1answer
942 views

Does the multiplicative identity have to be 1?

I am just starting out with vector spaces and I am having a hard time understanding them. One of the requirements states that $1\mathbf{v}=\mathbf{v}$ where $1$ is the multiplicative identity. Does 1 ...
6
votes
3answers
262 views

Are $\mathbb{C} \otimes _\mathbb{R} \mathbb{C}$ and $\mathbb{C} \otimes _\mathbb{C} \mathbb{C}$ isomorphic as $\mathbb{R}$-vector spaces?

Are $\mathbb{C} \otimes _\mathbb{R} \mathbb{C}$ and $\mathbb{C} \otimes _\mathbb{C} \mathbb{C}$ isomorphic as $\mathbb{R}$-vector spaces? I am having a very hard time at digesting tensor products ...
5
votes
1answer
146 views

Intuition for “the existence of a basis for every vector space is equivalent to the Axiom of Choice”?

Is there a intuitive way to understand "the existence of a basis for every vector space is equivalent to the Axiom of Choice"?
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4answers
141 views

Nullspace that spans $\mathbb{R}^n$?

My professor said that if for a $n \times n$ matrix $A$, $\text{null}(A) = \mathbb{R}^n$, then $A = 0_{n}$. Why is this true? I understand what its saying - if everything times this matrix is zero, ...
5
votes
1answer
195 views

Is there a difference between abstract vector spaces and vector spaces?

I am following my Oxford syllabus and my next step is abstract vector spaces, in my linear algebra book I've found vector spaces. I've searched a little and made a superficial comparison between ...
5
votes
4answers
127 views

Vector dimension of a set of functions

Let $F$ be a field and $S$ an infinite set. Set $V=\{f:S \rightarrow F\}$ endowed with the vector space structure that results from the pointwise operations of $F$. It is easy to prove that $|S| \leq ...
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4answers
613 views

Isomorphism of Vector spaces over $\mathbb{Q}$

From this post we see that $\mathbb{R}$ over $\mathbb{Q}$ is infinite dimensional. Similarly $\mathbb{C}$ over $\mathbb{Q}$ is also infinite dimensional, and I rememeber having solved a problem that ...
4
votes
1answer
114 views

If the expectation $\langle v,Mv \rangle$ of an operator is $0$ for all $v$ is the operator $0$?

I ran into this a while back and convinced my self that it was true for all finite dimensional vector spaces with complex coefficients. My question is to what extent could I trust this result in the ...
4
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1answer
142 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 ...
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votes
3answers
138 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 ...
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votes
3answers
362 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 ...
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votes
1answer
2k views

Dimension of Vector Space (Polynomial)

I was asked by a friend to: "Find the dimension of the vector space consisting of all polynomials in $n$-variables of degree at most $k$".Now, my response to him was that since the basis consists of ...
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2answers
3k views

Span of an empty set is the zero vector

I am reading Nering's book on Linear Algebra and in the section on vector spaces he makes the comment, "We also agree that the empty set spans the set consisting of the zero vector alone". Is Nering ...
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votes
2answers
1k views

Showing that a set of trigonometric functions is linearly independent over $\mathbb{R}$

I would like to determine under what conditions on $k$ the set $$ \begin{align} A = &\{1,\cos(t),\sin(t), \\ &\quad \cos(t(1+k)),\sin(t(1+k)),\cos(t(1−k)),\sin(t(1−k)), \\ &\quad ...
4
votes
1answer
207 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 $A$ ...
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votes
7answers
29k views

Finding a unit vector perpendicular to another vector

For example we have the vector $8i + 4j - 6k$, how can we find a unit vector perpendicular to this vector?
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3answers
22k views

Find the equation of the plane passing through a point and a vector orthogonal

I have come across this question that I need a tip for. Find the equation (general form) of the plane passing through the point $P(3,1,6)$ that is orthogonal to the vector $v=(1,7,-2)$. I would ...
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4answers
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Linear Algebra: determine whether the sets span the same subspace

So I am stuck on 51 here: 51. Determine whether the sets $S_1$ and $S_2$ span the same subspace of $\mathbb{R}^3$: $$\begin{align*} S_1 &= \Bigl\{ (1,2,-1),\ (0,1,1),\ (2,5,-1)\Bigr\}\\ ...
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1answer
365 views

What is the difference between Cartesian and Tensor product of two vector spaces

In particular, how is it that dimension of Cartesian product is a sum of dimensions of underlying vector spaces, while Tensor product, often defined as a quotient of Cartesian product, has dimension ...
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votes
2answers
163 views

What is (fundamentally) a coordinate system ?

Consider the following construction of vectors and points. Let's start with a vector space, or more specifically a coordinate space $F^N$ over a field $F$ and of $N$ dimensions. The elements of this ...
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votes
2answers
247 views

Easy visualizations of small countable ordinals

The ordinal number $\omega^2$ can be visualized as $\omega$-many copies of $\omega$. Likewise, the ordinal number $\omega^3$ can be visualized as $\omega^2$-many copies of $\omega$, arranged as ...
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2answers
144 views

Cross products?

Say you have vectors $v$ and $w$. Let there cross product be denoted by $x$ so that: $$v \times w = x$$ According to Wikipedia: $$x_x = v_yw_z - v_zw_y$$ $$x_y = v_zw_x - v_xw_z$$ $$x_z = v_xw_y - ...
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2answers
344 views

How to prove that two non-zero linear functionals defined on the same vector space and having the same null-space are proportional?

Let $f$ and $g$ be two non-zero linear functionals defined on a vector space $X$ such that the null-space of $f$ is equal to that of $g$. How to prove that $f$ and $g$ are proportional (i.e. one is a ...
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3answers
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Adding two subspaces

I have two subspaces: $$W_1 = \{(x, 3x) : x\in \Bbb R \}$$ and $$W_2 = \{(2x, 0): x\in \Bbb R \}$$ How do I get $W_1 + W_2$? I tried simply adding a sample vector from each, i.e. $$ (1, 3) + (2, ...
3
votes
1answer
437 views

If $X$ is infinite dimensional, all open sets in the $\sigma(X,X^{\ast})$ topology are unbounded.

As in the title, if $X$ is infinite dimensional, all open sets in the $\sigma(X,X^{\ast})$ topology are unbounded. The $\sigma(X,X^{\ast})$ topology is the weakest topology that makes linear ...
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2answers
452 views

short exact sequence of holomorphic vector bundles splits but not holomorphically, only $C^{\infty}$

If there is a short exact sequence of holomorphic vector bundles, $$0 \overset{a_1}{\to} W \overset{a_2}{\to} V \overset{a_3}{\to} F \overset{a_4}{\to} 0,$$ then one can expect a $C^{\infty}$ ...
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3answers
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Angle between two vectors?

I have been taught that the angle between two vectors is supposed to be their inner product. However, the book I'm reading states: Recall that the angle between two vectors $u = ...
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1answer
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Equation for non-orthogonal projection of a point onto two vectors representing the isometric axis?

Suppose I have two vectors that are not orthogonal (let's say, an isometric grid) representing the new axis. Suppose I want to project a point onto these two vectors, how would I do it? Dot product ...
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1answer
203 views

$4$ idempotent operators $E_1,E_2,E_3,E_4$ $:V\to V$ such that $E_1+E_2+E_3+E_4=I$ but don't partition the identity

Let $V$ be a vector space over $F$ such that $charF \neq 2$ Can anyone help me think of $4$ idempotent operators $E_1,E_2,E_3,E_4$ $:V\to V$ such that $E_1+E_2+E_3+E_4=I$ but $\{E_1,E_2,E_3,E_4 \}$ ...
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2answers
8k views

plane determined by 2 vectors

i have 2 perpendicular vectors in space . How can i determine the plane determined by the 2 vectors? Regards, Alexandru Badescu
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1answer
35 views

Does linear dependency have anything to do when determining a span?

Q: Does $\{(1,1) , (2,2)\}$ span $\mathbb{R}^2$? A: No, because they are linearly dependent. I agree that it doesn't span $\mathbb{R}^2$, but from my understanding, linear dependency has ...
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3answers
88 views

Matrix-free proof of $Z(GL_n(F)) = \{\lambda I:\lambda \in F^\times\}$?

How does one prove that $$Z(GL_n(F)) = \{\lambda I:\lambda \in F^\times\}$$ without resorting to matrices (and bases)? (BTW, $Z(GL_n(F))$ is the center of $GL_n(F)$, the general linear group of order ...
2
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1answer
346 views

Algebraic complements in vector space of functions without the axiom of choice

The axiom of choice is equivalent to the statement that every subspace $U$ of every vector space $V$ has an algebraic complement, i.e. another subspace $W$ that has a trivial intersection with the ...
2
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0answers
294 views

Horizontal and vertical tangent space of Orthogonal group

We know for the orthogonal group n-by-n orthogonal matrices, the tangents are given by $X^T\Delta + \Delta^TX = 0$ where $\Delta$ is the tangent. Now I was reading about the vertical and horizontal ...
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4answers
352 views

Proof: $\det\pmatrix{\langle v_i , v_j \rangle}\neq0$ $\iff \{v_1,\dots,v_n\}~\text{l.i.}$

Let $V$ be a real inner product space and $S=\{v_1,v_2, \dots, v_n\}\subset V$. How am I to prove that $S$ is linearly independent if and only if the determinant of the matrix $$ ...
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3answers
116 views

$I+A^*A$ is non-singular whenever $A$ is a square matrix with complex entries? [closed]

Let $A$ be a square matrix with complex entries , then is it true that $I+A^*A$ is non-singular ? where $A^*$ denotes the conjugate transpose of $A$ http://en.wikipedia.org/wiki/Conjugate_transpose ...