I needed to write a math song.
I wish this song had existed back in the 2nd year of uni.
The Vector Space Axioms
If you want to have a vector space over a field
If you want to get it tight
Then all the axioms must be in place
For it all to work out right
The axioms are simple, true
And they seem quite obvious
But if not strictly followed you
Might be confused and mussed!
The commutativity of addition - that's x + y is the same as y + x
The associativity of addition - that's x + y, plus z is the same as x, plus y +z
The existence of zero- an element in the space where x + 0 = x for all x in the field
The existence of the additive inverse - for every x, there is a y, so x+y = 0
For every x in the space, 1x = x
For every a and b in the field, a (b x) equals (ab) x
For every a in the field, a (x + y) equals ax + ay
For each pair a and b in the field (a+b)x = ax + bx
The axioms are simple, but it's true
Without these you just can't do
Linear algebra at all.