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 ...

learn more… | top users | synonyms

12
votes
3answers
4k views

Linear independence of functions

I want to determine whether 3 functions are linearly independent: \begin{align*} x_1(t) = 3 \\ x_2(t) = 3\sin^2(t) \\ x_3(t) = 4\cos^2(t) \end{align*} Definition of Linear Independence: $c_1x_1 + ...
10
votes
3answers
4k views

Question about basis and finite dimensional vector space

I have seen the statement "Every finite dimensional vector space has a basis." (Here on page 5) I'm confused about what this tells me. It seems to tell me nothing: by definition, the dimension of a ...
10
votes
3answers
334 views

What is the relationship between $(u\times v)\times w$ and $u\times(v\times w)$?

Given three vectors $u$, $v$, and $w$, $(u\times v)\times w\neq u\times(v\times w)$. This has been a stated fact in my recent class. But what is the ultimate relationship between them? I would presume ...
9
votes
1answer
142 views

What is needed to make Euclidean spaces isomorphic as groups?

Consider the abelian groups $G_n=(\mathbb R^n,+)$ for $n\geq1$. Claim: For any $n$ and $m$ the groups $G_n$ and $G_m$ are isomorphic. This claim is true if one assumes the axiom of choice, and I ...
8
votes
2answers
644 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, ...
7
votes
4answers
3k views

Linear algebra - Dimension theorem.

Suppose we have a vector space $V$, and $U$, $W$ subspaces of $V$. Dimension theorem states: $$ \dim(U+W)=\dim U+ \dim W - \dim (U\cap W).$$ My question is: Why is $U \cap W$ necessary in this ...
6
votes
1answer
10k views

Properties of a matrix whose row vectors are dependent

When a column vector in a matrix is a made up of "combination" of its other column vectors, it is said to be linearly dependant. Say... $$ A=\begin{bmatrix} 2 & 1 & 0\\ 4 & 5 & -6\...
6
votes
2answers
110 views

Does $\mathbb{R}^n$ have a real vector space structure with dimension other than $n$?

Can we define a vector space structure on $\mathbb {R}^n$ other than usual scalar multiplication and usual addition such that the dimension of $\mathbb {R}^n$ over $\mathbb {R}$ is not $n$ but some $m$...
4
votes
2answers
134 views

If $A$ is a complex matrix of size $n$ of finite order then is $A$ diagonalizable ?

Let $A$ be a complex matrix of size $n$ if for some positive integer $k$ , $A^k=I_n$ , then is $A$ diagonalizable ?
4
votes
1answer
1k views

Prove $\mathbb R$ vector space over $\mathbb Q$

I am proving that $\mathbb R$ is a vector space over $\mathbb Q$. So far, I have stated that vector addition and scalar multiplication trivially hold in $\mathbb R$. I then showed that $(\mathbb R, +)...
4
votes
2answers
240 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: $D_{\xi}(...
4
votes
1answer
86 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.
4
votes
1answer
630 views

An example of two infinite-dimensional vector spaces such that $\dim_{\mathbb{F}}\mathcal{L}(U,V)> \dim_{\mathbb{F}}U\cdot \dim_{\mathbb{F}}V$

I know that if $U$ and $V$ are vector spaces over a field $\mathbb{F}$ then $\dim_{\mathbb{F}}\mathcal{L}(U,V)\geq \dim_{\mathbb{F}}U\cdot \dim_{\mathbb{F}}V$. I am looking for an example of two ...
4
votes
2answers
704 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 ...
4
votes
1answer
211 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 ...
4
votes
1answer
3k 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 ...
4
votes
3answers
669 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 ...
3
votes
3answers
3k views

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 = (u_0,\ldots,u_{n−1}...
3
votes
2answers
436 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 ...
3
votes
3answers
227 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 $\operatorname{tr}(X^TDX)-2\operatorname{tr}(X^TDY)+\...
3
votes
3answers
160 views

determinant of the linear transformation $T(X) =\frac{1}{2} (AX+XA)$

Let $V$ vector space of all matrices $3\times3$, and let $A$ be the diagonal matrix : $$ \begin{pmatrix} 1 & 0 & 0\\ 0 & 2& 0 \\ 0 & 0& 1\end{pmatrix} $$ Compute thee ...
3
votes
4answers
6k views

Why does cross product tell us about clockwise or anti-clockwise rotation?

Wikipedia link for Cross Product it talks about using cross product to determine if 3 points are in clockwise or anti-clockwise rotation. I'm not able to visualize this or think of it in terms of math....
3
votes
3answers
845 views

Finding a basis for the solution space of a system of Diophantine equations

Let $m$, $n$, and $q$ be positive integers, with $m \ge n$. Let $\mathbf{A} \in \mathbb{Z}^{n \times m}_q$ be a matrix. Consider the following set: $S = \big\{ \mathbf{y} \in \mathbb{Z}^m \mid \...
3
votes
2answers
79 views

Alternative definition for span and proving it is equivalent to the most common one

This is a question related to something that I asked here about this alternative definition of span. User hardmath has helped me a lot! Therefore, I can't still understand how to prove the equivalence ...
2
votes
1answer
60 views

Maximizing and minimizing dot products

Given 2 vectors $u,v \in \mathbb{R^n}$ such that $\|u\| = 1$ and $\sum_{i=1}^n v_i= c$ where $c<1$, I would like to maximize $$\sum_{i=1}^n u_i v_i \log (v_i)$$ and minimize $$\sum_{i=1}^n u_i v_i \...
1
vote
1answer
356 views

Visualization of 2-dimensional function spaces

As a follow-up question to what is the norm measuring in function spaces I just had an idea: How about visualizing function spaces as normal planes. What I have in mind is to have an orthogonal ...
1
vote
2answers
1k views

Consistency of matrix norm: $||Ax||_2 \leq ||A||_{Frobenius}||x||_2$

I'm trying to show that $||Ax||_2 \leq ||A||_{F}||x||_2$ where $A$ is an n by n matrix, $x\in \mathbb R^n$, $||x||_2$ is the euclidean norm, and $||A||_F$ is the frobenius norm. I actually wrote ...
1
vote
1answer
107 views

Proof of $(u.v)=(u_{x}.v_{x}+u_{y}.v_{y}+u_{z}.v_{z})$, assuming $(u.v)=|u||v|cos\theta$.

I would really love any sort of proof of this. I have a very elementary geometric proof for $R^{2}$. That's mainly because I can easily represent $cos\theta$ in the form of $\frac{u_{x}v_{x}+u_{y}v_{...
1
vote
3answers
3k views

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 ...
1
vote
3answers
5k views

How to find an intersection of a 2 vector subspace?

Assuming we have 2 subspaces, $\mathbb W$ and $\mathbb U$ of $\mathbb V$. how to get thier intersection?
1
vote
3answers
103 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 ...
0
votes
1answer
475 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!
-1
votes
1answer
69 views

Proving that if two linear transformations are one-to-one and onto, then their composition is also.

I am attempting to solve a problem with the following given conditions: Let V, W. and Z be vector spaces, and let $T:V \longrightarrow W$ and $U: W\longrightarrow Z$ be linear.Prove that if U and T ...
14
votes
2answers
281 views

Infinite direct product of fields.

Let $F$ be a field, and consider the infinite direct product$$F \times F \times F \times F \times \dots,$$i.e. $\prod_{i=0}^\infty F$, i.e. the direct product of a countable number of copies of $F$. ...
14
votes
1answer
379 views

Do you need the Axiom of Choice to assert that every real vector space has a norm?

Math people: This question is 95% answered (the first answer) at Does every $\mathbb{R},\mathbb{C}$ vector space have a norm? and Vector Spaces and AC . The questions, answers, and links found there ...
10
votes
5answers
16k views

How to solve this to find the Null Space

What I did : I put this into reduced row echleon form: $$\begin{bmatrix} 1 & -2 & 2 & 4 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{...
9
votes
9answers
61k 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?
9
votes
6answers
1k 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 ...
8
votes
3answers
2k views

How can a subspace have a lower dimension than its parent space?

If $V$ is a vector subspace of $W$, then $$\dim(V) \le \dim(W)$$ Why? Does that mean that for $$W = \mathbb{R}^3\\ V = \{(0,0)\}$$ $V$ is a valid subspace of $W$? But $V$ only has two ...
7
votes
1answer
2k 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
2answers
296 views

How do we show every linear transformation which is not bijective is the difference of bijective linear transforms?

I have been reviewing some ideas about vector spaces and came upon a surprising fact. I am not quite sure how to begin the argument because the problem requires one to construct two bijective linear ...
5
votes
2answers
581 views

Sparse basis for linear subspace

Suppose I have a linear subspace of some vector space, e.g. described as the column space of some big matrix. How would I algorithmically find a basis of that same subspace where the basis matrix is ...
5
votes
1answer
841 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 ...
5
votes
4answers
136 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 ...
5
votes
3answers
3k views

Prove that Every Vector Space Has a Basis

My textbook extended the following proof to show that every vector space, including the infinite-dimensional case, has a basis. Condition: $S$ is a linearly independent subset of a vector space $...
5
votes
2answers
103 views

Confused about Euclidean Norm

I am trying to understand that the Euclidean norm $\|x\|_2 = \left(\sum|x_i|^2\right)^{1/2}$ is in fact a norm and having trouble with the triangle inequality. All the proofs I have referred to ...
5
votes
4answers
155 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
170 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"?
4
votes
3answers
154 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
votes
1answer
2k views

Grassman formula for vector space dimensions

If $U$ and $W$ are subspaces of a finite dimensional vector space, $$ \dim U + \dim W = \dim(U\cap W) + \dim(U + W)$$ Proof: let $B_{U\cap W} = \{v_1,\ldots,v_m\}$ be a base of $U\cap W$. If we ...