How and why vector spaces are defined?

Question

Is it true that vector spaces are defined to check if system of linear equations is solvable or not?

Explanation: Goal is to solve system of linear equations.

In matrix form: $Ax = b$. As $A = [C_1 \; C_2 \; ... \; C_n]$, where $C_n$ is a column and $x = [x_1 \; x_2 \; ... \; x_n]$.

Therefore, $C_1x_1 + C_2x_2 + ....+ C_nx_n = b$. Linear combination of column vectors produce vector $b$.

Because of above statement (linear combination) we choose a set of vectors that have closure under addition and scalar multiplication (closure under linear combination) and call that set of vectors a vector space. Now, if vector $b$ lies in that set of vectors (vector space) then only system of linear equations is solvable.

Answer 1

The resolution of the systems of linear equations is one of the main goal but of course there are many others applications (i.e. least squares, dynamical systems, image compression, etc,.).

See also the related Why study linear algebra?

Answer 2

Please note that the notion of vector spaces and more generally that of $R$- Module is fundamental in maths about the last 200 years.

Answer 3

Vector spaces are the appropriate context for studying linear transformations. A transformation $T$ is linear if its maps a linear combination of two elements from its domain to a linear combination of the individual images of the two elements.

More formally $T$ is linear if $T(p \text{a} + q\text{b})) = pT(\text{a} + qT(\text{b})$. Less formally, we can say that $T$ is linear if its preserves straight lines i.e. the image of a straight line in its domoan is always a straight line in its range.

Vector spaces introduce just the right amount of structure to allow us to define and study concepts such as straight lines, linear combinations, linear transformations and linear equations. They do not introduce any unnecessary structure such as distances, angles, limits, differentiation/integration etc.

Many physical phenomena are represented (or at least approximated) by linear models. So vector spaces are important in the physical sciences as well as being interesting in themselves.

Answer 4

Several answers to the question in your title, if not in the body, is here: How did mathematicians decide on the axioms of linear algebra.