Vector Spaces: Explained (2x2 Matrices)

[das1]

Can someone explain this to me? Thanks!

The component in the ith row and jth column of a matrix can be
labeled m(i,j).
In this sense a matrix is a function of a pair of integers.
For what set S is the set of 2 × 2 matrices the same as the set R^s ?

Generalize to other size matrices.

 

[Deveno]

Well a 2x2 matrix has 4 numbers you need to specify.

How many (real) numbers to you need to specify in $\Bbb R^n$?

What they are trying to get you to see, is that a vector $(v_j) \in \Bbb R^n$ can be thought of as a function:

$f: \{1,2,\dots,n\} \to \Bbb R$, with:

$f(j) = v_j$.

For example, we can identify the vector $(1,3,-2) \in \Bbb R^3$ with the function:

$1 \mapsto 1$
$2 \mapsto 3$
$3 \mapsto -2$

the "order" of the coordinates is determined by the "natural" order of the natural numbers 1,2,3.

It may help to recall that the number of pairs from two finite sets $S,T$ is:

$|S \times T| = |S|\ast |T|$.

With $S = \{1,2\} = T$, this gives 4 pairs:

(1,1),(1,2),(2,1),(2,2)---the same number of possible "indices" of a 2x2 matrix.

EDIT: the set $S$ this problem is looking for might be "any" set with the proper cardinality (size). However, for finite sets, it is common to use for a set of cardinality $k$, the set $\{1,2,\dots,k\}$, which certainly has $k$ elements, because there is a "natural" way to describe:

1st element, 2nd element, etc. for this set.