if you google free linear algebra books you will find a lot of them.. there is even one or two or thre on my web page.
Basically linear algebra is about objects "vectors" that can be added and scaled by numbers, and operations that take sums to sums and scale vectors to vectors scaled by the same number.
All su8ch linear operations (in finite dimensiopnal spaces) can be represented as a` type of multiplication, which is a sequence of dot products, so that the operation acts like
X goes to AX, for some "matrix" A of numbers,
then we want to know when the equation AX = B can be solved for X, and if so how many solutioins there are.the basic result is that this depends on counting dimensions.
I.e. the fundamental theorems says that if the "null space" is the solutions of AX=0,
then the space of B's that can be solved for has dimension = the dimension of the X's minus the dimension of the nullspace.
So if we are looking at an operation X goes to AX, where the X's have dimension 9 and the B's have dimension 5, and the null space has dimension 4, then in fact every B can be solved for in the equation AX=B.
A sequence of vectors v1,...,vr in the null space say, "spans that null space, if every vector in the nullspace can be written in terms of these guys using just addition and scaling.If also no one of the vj can be written in terms of the other vi's, then the sequence v1,...,vr is "independent" and the dimension of the null space it spans is equal to the number of these guys, i.e. to r.So to fimd dimensions of spaces you find an independent spanning set, called a "basis" and then count the number of vectors in it.To obtain such a basis you try to find a spanning set and then eliminate dependent vectors from it. This can be done with numerical vector by a mechanical process called gauss elimination.If the space of B's has dimension 8 say, and the spoace of X's has dimension 10 say, then the null space of those X's with AX=0 must have dimension at least 2. If it has dimension exactly 2 then the equaiton AX=B has a solution for every B, and the solution set is always also of dimension 2 (equal to the size of the null space).
If the null space is bigger, say dimension 5, then the space of B's that can be solved for in AX=B is only of dimension 10-5 = 5, so most B's cannot be solved for. But if a particular B can be solved for as AX=B for some, then the space of X's that solve it always has the same dimension as the null space, namely 5 in this case.
