I have a set of 4 vectors (a1,a2,a3,a4) in R3 and a set of 4 vectors (b1,b2,b3,b4)in R4. I need to show that there is precisely one linear map f: R3 --> R4 with f(ai) = bi

a1 = (1,0,0)

a2= (0,1,0)

a3 = (0,0,1)

a4 = (2,1,3)

b1 = (1,2,4,1)

b2 = (1,1,0,1)

b3=(-1,0,4,-1)

b4 = (0,5,20,0)

I have found the linear map (x+y-z,2x+y,4x+4z,x+y-z)

how would I show that this is unique? I then need to find the kernel and the image of f.. would the image just be (x+y-z,2x+y,4x+4z,x+y-z)

and the kernel would be (1,-2,-1)^T ?

many thanks