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a matrix is a transformation of a vector from one vector space to another through a linear transformation.

lets say we are going from ℝ

^{3}to ℝ

^{2}

f(v) = f(vx+vy+vz) = f(vx) + f(vy) + f(vz)

This is a linear transformation so these properties can be exploited.

Now we must define how the basis vectors of a vector in a certain vector space are transformed.

The basis vectors can be written as a linear combination of the basis vectors of the vector space as well.

vx = 1vx + 0vy + 0vz

vy = 0vx + 1vy + 0vz

vz = 0vx + 0vy + 1vz

The linear transformation transforms these basis vectors into a linear combination of these basis vectors.

f(vx) = 2w1 + 5w2

f(vy) = 3w1 + 7w2

f(vz) = 4w1

Finally my question is how would you write that if as a coordinate function f(vx, vy, vz) = ?

That equation would then have to satisify all the basis vectors as well

f(1,0,0) = ?

f(0,1,0) = ?

f(0,0,1) = ?

so when you add all these values together you should get the same as a matrix multiplication of

[2 3 4] *[1]

[5 7 0] [1]

[1]

I have no idea how to write it because i have rarely worked with coordinate functions and cannot find much on it online. But i have come to realize that the key to understanding matrices is to understand coordinate functions. Thanks alot