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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Spans of Orthogonal complements

Let $A$ be the matrix $$ \begin{pmatrix} 1 & 1 & -1&-1 \\ 1 & 2 & -2 & 1 \\ \end{pmatrix} ,$$ let $W$ = ker $A$ and let $W^\bot$ be the ...
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Find matrix of linear transformation relative to new bases

If $T:\Bbb R^3\to \Bbb R^2$ is a linear transformation, and the matrix of $T$ = $\left(\begin{array}{ccc}0 & 1 & 1 \\0 & 1 & -1\end{array}\right)$. If you use the basis $\{i,j,k\}$ ...
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Vector spaces and infinite cyclic linear transformations II

Let $V$ be the vector space of infinite dimension on the field $\mathbb{Z}_2$. Let's say $$ V=\langle v_1\rangle+\langle v_2\rangle+\dots+ \langle v_n\rangle+\dots$$ where each $v_i$ has order $2$ and ...
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59 views

Given a field extension $K\colon F$, $K$ is an $F$-vector space

I'm having a hard time understanding fields. Could someone help with the following I need to show that if $F$ $\subseteq$ $K$ are both fields and addition and multiplication on F are the ...
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Determining a basis of vector space with unusual addition and multiplications: $(a,b)+(c,d)=(ad+bc,bd) $ and $k * (a,b) =(kab^{k-1},b^k)$

Let $\mathbb W=\{(a,b) \in \mathbb R^2 \mid b>0\}$ and define addition by $(a,b)+(c,d)=(ad+bc,bd) $ and define scalar multiplication by $k * (a,b) =(kab^{k-1},b^k)$. Find a basis for $\mathbb ...
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71 views

Having trouble with a linear lager question about kernel and basis

Let $V$ be a $2$-dimensional vector space, and let $\alpha={e_1,e_2} $ be a basis for $V$. Define a linear transformation $T: V\to V$ by declaring that: $T(e_1+e_2)=2e_1−e_2 $ $T(e_2)=4e_1−2e_2$. ...
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matrices forms a basis for vector space 2x2

$\begin{bmatrix}0&1\\2&3\end{bmatrix}$ $\begin{bmatrix}3&4\\5&6\end{bmatrix}$ $\begin{bmatrix}7&8\\9&10\end{bmatrix}$ $\begin{bmatrix}11&12\\13&14\end{bmatrix}$ Show ...
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1answer
100 views

Find where $r(t)=<t,t,t^2>$ hits the $x-y$ plane

I have to find $r'(t)$ and $||r'(t)||$ for $r(t)=<t,t,t^2>$, which I know how to do. $r'(t)=<1,1,2t>$ $||r'(t)||=\sqrt{2+4t^2}$ The problem is that my professor didn't explain how to ...
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Proof help: The span of any list of vectors in V is a subspace of V.

I'm having some trouble understanding this proof. Let $U = ()$, the empty set, and define span$(U) = 0 \subset V$. Now let $U = (v_1 \cdot \cdot \cdot v_n)$ be a list of vectors in $V$. This means ...
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263 views

How do you prove $\def\rank{\operatorname{rank}}\rank(f_3 \circ f_2) + \rank(f_2 \circ f_1) \leq \rank(f_3 \circ f_2 \circ f_1) + \rank(f_2) $? [duplicate]

Possible Duplicate: Lower bound involving the rank of the composition of linear transformations The following question is about a lower bound on the rank of a composition of functions given ...
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2answers
333 views

Lower bound involving the rank of the composition of linear transformations

The following question is about a lower bound on the rank of a composition of functions given as a simple expression for the two terms of the sum involved in the inequality. Consider ...
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4answers
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Given two basis sets for a finite Hilbert space, does an unbiased vector exist?

Let $\{A_n\}$ and $\{B_n\}$ be two bases for an $N$-dimensional Hilbert space. Does there exist a unit vector $V$ such that: $$(V\cdot A_j)\;(A_j\cdot V) = (V\cdot B_j)\;(B_j\cdot V) = 1/N\;\;\; \ ...
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1answer
692 views

Why is the inclusion of the tensor product of the duals into the dual of the tensor product not an isomorphism?

Let $V$ and $W$ be vector spaces (say over the reals). There is a linear injection $V^* \otimes W^* \to (V \otimes W)^*$ which sends $\sum_i f_i \otimes g_i \in V^* \otimes W^*$ to the unique ...
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3answers
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How to tell if a set of vectors spans a space?

I want to know if the set $\{(1, 1, 1), (3, 2, 1), (1, 1, 0), (1, 0, 0)\}$ spans $\mathbb{R}^3$. I know that if it spans $\mathbb{R}^3$, then for any $x, y, z, \in \mathbb{R}$, there exist $c_1, c_2, ...
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3answers
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What is an inner product space?

As I've understood it, what I've learned is that the dot product is just one of many possible "inner product spaces". Can someone explain this concept? When is it useful to define it as something ...
12
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1answer
627 views

Why it is important for isomorphism between vector space and its double dual space to be natural?

I'm reading the book (by A. Kostrikin) on linear algebra and I feel like I'm really missing something about this idea. I understand the formal proofs of: a) isomorphism between vector space $V$ and ...
9
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1answer
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Vector, Hilbert, Banach, Sobolev spaces

Trying to wrap my head around all these different spaces. Which one is the most general? Can you summarize the differences between them? Is there a notable space that I missed?
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*understanding* covariance vs. contravariance & raising / lowering

There are lots of articles, all over the place about the distinction between covariant vectors and contravariant vectors - after struggling through many of them, I think I'm starting to get the idea. ...
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1answer
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Effect of elementary row operations on determinant?

1) Switching two rows or columns causes the determinant to switch sign 2) Adding a multiple of one row to another causes the determinant to remain the same 3) Multiplying a row as a constant results ...
15
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1answer
993 views

Prove $\mathbb{Z}$ is not a vector space over a field

This is an exercise from Chapter 3 of Golan's linear algebra book. Problem: Show $\mathbb{Z}$ is not a vector space over a field. Solution attempt: Suppose there is a such a field and proceed by ...
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How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$?

How to show that quotient space $X/Y$ is complete when $X$ is Banach space, and $Y$ is a closed subspace of $X$? Here's my attempt: Given a Cauchy sequence $\{q_n\}_{n \in \mathbb{N}}$ in $X/Y$, each ...
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5answers
707 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 ...
5
votes
3answers
434 views

Why a non-diagonalizable matrix can be approximated by an infinite sequence of diagonalizable matrices?

It is known that any non-diagonalizable matrix, $A$, can be approximated by a set of diagonalizable matrices, e.g. $A \simeq \lim_{k \rightarrow \infty} A_k$. Why this is true? Note: I was faced with ...
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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 ...
5
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3answers
496 views

Why do we use n-dimensional spaces?

On mathoverflow, Terry Tao says the following: For instance, one can view a high-dimensional vector space as a state space for a system with many degrees of freedom. A megapixel image, for instance, ...
10
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3answers
260 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 ...
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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 ...
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2answers
151 views

Does $(x,f(x),\cdots,f^p(x))$ is linearly dependent over $E$ implies $(id, f, …, f ^ p)$ is linearly dependent over $\mathcal{L}(E)$?

Here is the original (classic I think) problem I had encored: if $(x,f(x))$ is a linearly dependent family of $E$ (a vector space) for all $x\in E$, then the family $(id,f)$ is linearly dependentt ...
8
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2answers
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Can a basis for a vector space be made up of matrices instead of vectors?

I'm sorry if this is a silly question. I'm new to the notion of bases and all the examples I've dealt with before have involved sets of vectors containing real numbers. This has led me to assume that ...
6
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4answers
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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
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2answers
263 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
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1answer
231 views

Cross Product Intuition

I know the cross product between a vector $a$ and a vector $b$ is just a vector whose magnitude is the product of magnitude of $b$ times the magnitude of the perpendicular component of $a$ in relation ...
5
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1answer
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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 ...
5
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0answers
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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 ...
5
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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 ...
4
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2answers
52 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
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2answers
190 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
52 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 ...
3
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2answers
51 views

Function for diagonalizing a vector.

I was playing around whith the idea of what operation (function) should I perform (apply) over a vector $\mathbf{a} = (a_1,a_2, \ldots, a_N)^T \in \mathbb{R}^N$ to come up with the following matrix: ...
3
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2answers
210 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
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188 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
72 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.
3
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5answers
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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 ...
3
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2answers
434 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 ...
3
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3answers
209 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 ...
2
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3answers
2k views

Vectors that form a triangle!

I have a problem here. How can I prove that sum of vectors that form a triangle is equal to 0 $(\vec {AB}+\vec {BC}+\vec {CA}=\vec 0)$ ? Thank you!
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3answers
93 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
98 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 ...
0
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1answer
431 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
2k 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 ...