Tagged Questions
0
votes
0answers
32 views
What is a parallel vector space and how do I show it is isomorphic to the solution space?
How can I create an isomorphism between the solution space and a parallel vector space. I'm not sure how to define the vector space and the isomorphism.
$$ \begin{bmatrix}
-2 & 4 \\
...
0
votes
2answers
47 views
Difference between Spanning set and Postitive Spanning Set
I do understand the difference as mentioned in the texts about spanning set and positive spanning set, im somehow missing how if $v_1.. v_r$ is a positive spanning set for $R^n$, then $v_2 ... v_r$ ...
4
votes
1answer
42 views
Can you define a vector space in terms of a pre-existing projective space?
Projective spaces are usually defined as the quotient of a vector space (by the equivalence relation that identifies collinear vectors). However, in my opinion, projective spaces seem intuitively less ...
3
votes
3answers
52 views
Definition of a tensor field
Could anybody explain to me the following:
If $$T_{ij}=\nabla_i V_j-\nabla_j V_i$$
where $V_i$ is a covector field and $\nabla_i$ is the covariant derivative,
then $T_{ij}$ is a tensor field.
...
1
vote
1answer
238 views
Why are vector spaces sometimes called linear spaces?
I have never come across the term 'linear space' as a synonym for 'vector space' and it seems from the book I am using (Linear Algebra by Kostrikin and Manin) that the term linear space is more ...
1
vote
1answer
53 views
Suppose I said “$X$ spans $W$”…
So I've seen two definitions of this:
Let $V$ be a vector space with subspace $W$. We say that $X \subseteq V$ spans $W$ if and only if
(Definition 1): Every $\vec{w} \in W$ can be written as ...
0
votes
1answer
62 views
Question about piecewise linear paths
I'm having trouble with a concept about piecewise linear paths. I've been looking on the net but I haven't found a definition of it.
Consider a finite family of hyperplanes in a finite-dimension real ...
9
votes
3answers
793 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 ...
1
vote
5answers
159 views
Why is there implied an equality between vectors and $n$-tuples?
Are they considered equal in some sense?
For instance, I always write "...for vector $\mathbf{x} \in {\mathbb{R}}^n$ we have ...". I have a small problem with this (not a big one). The problem comes ...
1
vote
2answers
25 views
Definition clarifications on functions of vectors and their derivatives
Definition clarifications would be appreciated:
How do I interpret the following ? For $f: R^n\to R^m$, $Df(\vec\xi)(\vec{x})$ in differentiation of a vector function. I know it as a function that ...
1
vote
2answers
132 views
exterior product definition
i have question from vector mathematics,i know that if there is given two vector, for instance $a=\{a_1,a_2,a_3\}$,$b={b_1,b_2,b_3}$; then so called exterior product is determined as $a\wedge ...
1
vote
2answers
280 views
Is this Vector operation defined? Does it have a name?
Let's say I have 2 vectors:
[a, b, c]
[x, y, z]
And I need to do an operation like the following for a computer program:
...
