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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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
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1answer
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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 ...
4
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1answer
608 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
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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
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1answer
246 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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3answers
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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 ...
3
votes
2answers
147 views

On the canonical isomorphism between $V$ and $V^{**}$

I am trying to understand more about the Bidualspace (or double dual space). The whole idea is that $V$ and $V^{**}$ are canonically isomorphic to one another, which means that they are isomorphic ...
3
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2answers
293 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 ...
3
votes
1answer
732 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, ...
3
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2answers
764 views

Unique complement of a subspace

Can we have a unique complement to a subspace of a finite dimensional vector space V (exclude the trivial cases 0 and V) ?
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3answers
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For which values of a do the following vectors for a linearly dependent set in $R^3$?

For which values of a do the following vectors for a linearly dependent set in $R^3$? $$V_1= \left(a,\, \frac{-1}{2}, \,\frac{-1}{2}\right),\;\; V_2= \left(\frac{-1}{2},\, a, \,\frac{-1}{2}\right),\; ...
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votes
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
812 views

How many different basis' exist for an n-dimensional vector space in mod 2?

Imagine a 2-dimensional vector space in $\mathbb{Z} /2 \mathbb{Z}$. The only possible basis' are $$ \left(\begin{array}{c} 1\\ 0 \end{array}\right), \left(\begin{array}{c} 0\\ 1 ...
3
votes
1answer
224 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 $\{E_1,E_2,E_3,E_4 \}$ ...
2
votes
3answers
52 views

Eigenvalues of matrix with all $1$'s. [on hold]

Let $A$ be the $n \times n$ matrix over a field of characteristic 0, all of whose entries are 1. What are the eigenvalues of $A$, counted with their multiplicities?
2
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1answer
39 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 ...
2
votes
3answers
96 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
votes
0answers
364 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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votes
3answers
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Dual space and inner/scalar product space

$V$ is vector space of finite dimension. $〈· , ·〉$ is an inner product on $V$.(Field $F$) We set transformation $T \colon V \rightarrow V^*$ as the following: $(T(v))(w) = 〈v , w〉$. Prove that $T$ ...
2
votes
3answers
792 views

Induced Exact Sequence of Dual Spaces

So given a short exact sequence of vector spaces $$0\longrightarrow U\longrightarrow V \longrightarrow W\longrightarrow 0$$ With linear transformations $S$ and $T$ from left to right in the ...
2
votes
2answers
574 views

Norm-preserving map is linear

How can one show that a norm-preserving map $T: X \rightarrow X'$ where $X,X'$ are vector spaces and $T(0) = 0$ is linear? Thanks in advance.
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0answers
58 views

How to calculate the dimension of an infinite direct product of copies of a field?

Let $F$ be a field and $I$ an arbitrary infinite index set. I'd like to know how to calculate the dimension of $\prod_{i\in I}F$. By the way, I know $\dim(\prod_{i\in I}F)\geqslant ...
1
vote
1answer
78 views

Calculating a spread of $m$ vectors in an $n$-dimensional space

My question is regarding spreading $m$ vectors in an $n$ dimensional space such that the vectors are maximally distant from each other. For example, let us say I have a 2-D space, and 3 vectors, the ...
1
vote
1answer
87 views

Example for finite dimensional analog of integral transforms

I understand that integral transforms are generalisations of the dot product of functions that could be interpreted as infinite dimensional vectors. The most significant advantage then is that ...
1
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1answer
73 views

Farey Sequence Vector Orthogonality Relation Question

Take the Farey sequence $\mathcal{F}_n$ for $n=39$ with values $a_m\in \mathcal{F}_n$ and put them into a vector $$ \vec v_k=\biggr(\exp(2\pi i k a_m)\biggr)_m $$ Since Merten's function for $n=39$ ...
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2answers
122 views

Vector space of real numbers over the rational numbers

What is the easiest way to show that $\mathbb R$ is not finitely generated over $\mathbb Q$ ?
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2answers
179 views

What is the product and coproduct of Morphism category(Arrow category)?

Given category C, Its morphism category D means a category that has 1) "morphisms of C" as its objects 2) "pair (f,g) s.t. the diagram(square) commutes" as its morphisms The above definition is ...
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3answers
88 views

Dependency of linear map definition on basis

The definition of linear map depends on the basis. Is this a flaw of the "linear map" construct?
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3answers
127 views

What's special about the first vector

My linear algebra notes state the following lemma: If $(v_1, ...,v_m)$ is linearly dependent in $V$ and $v_1 \neq 0$ then there exists $j \in \{2,...,m\}$ such that $v_j \in span(v_1,...,v_{j-1})$ ...
1
vote
1answer
174 views

Dimensions of vector subspaces

Given a bilinear map $B:X\times Y\to F$ where $X,Y$ are vector spaces and given $S\leq X$, why is $\dim S+\dim \operatorname{ann}(S)=\dim Y$ where $\operatorname{ann}(S)$ is the annihilator of $S$ ...
1
vote
1answer
299 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 ...
0
votes
2answers
194 views

Partial Derivative of a outer product in Vector Calculus

I am trying to compute the partial derivative of certain vector products for calculating the stiffness matrix. So we already know that For any vector $\textbf{x}$, we have 1) The derivative of the ...
0
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0answers
22 views

Searching for a definition for n-Dimensional rotation which is cosine-distance invariant

I am wondering if it is possible to define a rotation for an $n$-Dimensional space ($n=2,3,4,5,\dots$). Given any vector $\vec v$, and knowing that it should be rotated to ...
0
votes
3answers
130 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 ...
0
votes
1answer
122 views

Finding loci of possible points satisfying vector simultaneous equations

I recently had an exam and a question came up which I was only partially able to answer. The question was the following: Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be constant vectors in ...
0
votes
1answer
163 views

inner product space , dual space, proof about isomorphism

Let $V$ be a vector space (not necessary being finite dimensional) and let $U,W$ be subspaces of $V$ such that $V = U\oplus W$. Prove that $V^\ast/(W^0)$ is isomorphic to $W^\ast$. Notation and ...
0
votes
1answer
88 views

V = U⊕W then Prove that (V/W)* is isomorphic to W^0

Let $V$ be a vector space (not necessary being finite dimensional) and let $U$, $W$ be subspaces of $V$ such that $V = U\oplus W$. Prove that $(V/W)^*$ is isomorphic to $W^0$. note: (V/W)* is the ...
0
votes
3answers
107 views

Linear Algebra: Matrix Spanning/Consistency Question

1) If there are $5$ vectors found in $\mathbb{R}^7$ will these vectors Span $\mathbb{R}^7$? Please explain. 2) Give an example of a $3$ by $5$ matrix for which all systems, $Ax=b$ for any $b$ in ...
0
votes
4answers
75 views

A vector should more be thought an identity of an entity in space rathar than magnitude + direction?

Can I say that vector is more like a "unique identity" of an entity in space rather than calling it an entity with magnitude and direction ? For example a line. A vector $(10,10,0)$ is the identity ...
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votes
0answers
101 views

Finding a Vector not in a Collection of Proper Subspaces

Let's take $K$ to be an infinite field and $V$ a vector space over it. Allowing $U_1,...U_l$ to be $l$ proper subspaces of $V$ (none the same), and further assuming that for all $i$, $U_i$ is not in ...
0
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1answer
327 views

How is it possible for the inverse function of a linear-continuous-bijective function to be not continuous?

If $E$ and $F$ are two normed vector spaces, $f:E\rightarrow F$ is a linear-continuous-bijective function. Then naturally I would think that $f^{-1}$ is also linear-continuous-bijective. But the ...
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votes
2answers
64 views

Some question about extension of bounded linear operator

Let we have the following bounded linear operator $$T: D(T)\rightarrow Y$$ such that $D(T)$ is the domain and it is a vector space and $Y$ is a Banach space . Then it has an extension $$H: ...
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votes
2answers
114 views

Definition of dimension

Let us consider Euclidean space $\mathbb{R}^n$. We say it is $n$-dimensional because each vector in it is an $n$-tuple $(x_1,...,x_n)$. However, it is possible to represent this exact same space using ...
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5answers
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What are some examples of infinite dimensional vector spaces?

I would like to have some examples of infinite dimensional vector spaces that help me to break my habit of thinking of $R^n$ when thinking about vector spaces.
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2answers
232 views

If $V \times W$ with the product norm is complete, must $V$ and $W$ be complete?

Let $V,W$ be two normed vector spaces (over a field $K$). Then their product $V \times W$ with the norm $\|(x,y)\| = \|x\|_V + \|y\|_W$ is a normed space. Using this norm it's easy to show that if ...
8
votes
2answers
242 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, ...
8
votes
5answers
3k 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 ...
8
votes
2answers
236 views

Can a basis for a vector space $V$ can be restricted to a basis for any subspace $W$?

I don't understand why this statement is wrong: $V$ is a vector space, and $W$ is a subspace of $V$. $K$ is a basis of $V$. We can manage to find a subset of $K$ that will be a basis of $W$. ...
8
votes
3answers
469 views

Why define vector spaces over fields instead of a PID?

In my few years of studying abstract algebra I've always seen vector spaces over fields, rather than other weaker structures. What are the differences of having a vector space (or whatever the ...
6
votes
1answer
83 views

Invariant Subspace of Two Operators [duplicate]

Let $S$, $T$ be linear operators on a finite-dimensional vector space $V$ over $\mathbb{C}$. Suppose $$S^2 = T^2 = I.$$ Show that there exists either a $1$-dimensional or $2$-dimensional ...