1
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1answer
32 views

A question on the rectangular region defined for a vector in $\mathbb{R}^N$

Let $K = (k_1,k_2,k_3,...k_N)$ be a vector in $\mathbb{R}^N$, consider the region $S_K$ consisting of all vectors $L = (l_1,l_2,l_3,...l_N)$ such that, $|l_i| \le |k_i| \forall i \in \{1,2,3,...N\}$. ...
0
votes
1answer
36 views

Vectors-Can anyone explain me the concept of sense in vectors?

Is it same as the direction? Then, why another term "sense"is used, instead of direction? Can anyone illustrate it?
0
votes
0answers
88 views

What is the operation inverse to vectorization (vec operator)?

There is a well knows vectorization operation in matrix analysis $\mbox{vec}$: https://en.wikipedia.org/wiki/Vectorization_%28mathematics%29 I've vectorized my matrix equations, did some ...
3
votes
2answers
53 views

Is there a name for the set of bit combinations of bitstrings?

Let $A \subset \{0,1\}^n$ be a set of $n$-bit bit vectors. Let me call a bit vector $b = (b^{(1)}, b^{(2)}, \dotsc, b^{(n)}) \in \{0,1\}^n$ a "bit combination" of the vectors in $A$ if: $$\forall i ...
1
vote
2answers
63 views

What does the term “distinguished basis” mean?

I know what a basis is (talking about vector spaces here), but I don't know what a distinguished basis is. Can you please explain the difference to me? I did not grow up in an English-speaking ...
1
vote
1answer
96 views

What does it mean when a set is said to be a “finitely generated vector space”?

I've somehow managed to go through 3 years of my Maths degree without truly understanding what the term "finitely generated vector space" means. Generated by what? Generating what? And for what? I ...
0
votes
1answer
40 views

Terminology for a weaked vector space

Let $S$ be a semigroup on which acts $\mathbb{R}_{\geq0}$. Does this structure has a name? For example $S$ can be the set of convex bodies in $\mathbb{R}^n$ with the Minkowsky sum.
5
votes
1answer
187 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 ...
2
votes
1answer
52 views

Could a subspace of a normed linear space be not a linear subspace?

On page 38, Functional Analysis, Pater Lax: Let $X$ be a normed linear space, $Y$ a subspace of $X$, The closure of $Y$ is a linear subspace of $X$. But on Wikipedia, linear subspace A ...
0
votes
1answer
65 views

What is a “rotated” basis?

My text (p. 19) introduces the concept of a "rotated" basis without explanation. What properties or characteristics of a basis make it "rotated" with respect to another? What operation on one basis ...
4
votes
1answer
2k 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 ...
2
votes
0answers
95 views

Simple linear operator?

From Wikipedia a linear operator T on a finite-dimensional vector space is semi-simple if every T-invariant subspace has a complementary T-invariant subspace. I wonder if there is a concept for ...
3
votes
2answers
1k views

Relation between Interior Product, Inner Product, Exterior Product, Outer Product..

Following my previous question Relation between cross-product and outer product where I learnt that the Exterior Product generalises the Cross Product whereas the Inner Product generalises the Dot ...
5
votes
1answer
2k views

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 ...
4
votes
1answer
189 views

Is there a name for this $k$-fold vector product?

Let $V$ be a set of vectors of length $n$. Define a $k$-fold product on $V$, $$ \Upsilon(\{v_1,\ldots,v_k\}):=\sum_{j=1}^n\prod_{i=1}^k v_{ij}, $$ where $v_i\in V$ and $v_{ij}$ is the $j^\text{th}$ ...
1
vote
1answer
166 views

What is an honest basis?

In a comment to this question, the commentator stated that "the monomials form an honest basis for your vector space". To be honest, I never heard of that. Is this something elementary?
0
votes
3answers
179 views

What do I call a unit vector parallel to a coordinate axis?

What do I call an arbitrary element of this set of vectors? $$ \begin{align*} \{&\langle 1, 0, 0 \rangle, \\ &\langle 0, 1, 0 \rangle, \\ &\langle 0, 0, 1 \rangle, \\ &\langle ...
1
vote
1answer
95 views

What is the proper term for the entity that relates a vector space and a set?

One way to generate a metric for a set $S$ (a distance function between elements $a,b$ of the set $S$) would be by associating it with a vector space $V$ (the vectors that connect the elements $a,b$) ...
1
vote
0answers
30 views

Name this concept: Comparing equal sized vectors vs. comparing features

If you obtain a vector by taking $n$ discrete samples over some underlying function, then it's easy to compare that vector with another of the same size. With a bunch of $n$-dimensional vectors, you ...
6
votes
1answer
245 views

What is the term for the projection of a vector onto the unit cube?

Normalizing a vector sets its magnitude to $1$ and retains its direction. In three dimensions, it projects the vector onto the unit sphere. Is there a term associated with projecting it onto the ...
1
vote
1answer
165 views

Is there any math operation defined to obtain vector $[4,3,2,1]$ from $[1,2,3,4]$?

I mean have it been studied, does it have a name? Like Transpose, Inverse, etc.. have names. I wonder if the "inversion" of the components position have a name so then I could search material on ...
1
vote
1answer
72 views

Dimension and size of an array, matrix, vector

For a $1 \times n$ or $n \times 1$ vector, I remember people say it is n-dimensional. For a $n \times m$ matrix, I heard it is said to have size $n \times m$. As to its dimension, quoted from ...