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# Vector spaces Watch

1. Define a function f : R3 --> R2 by

f(x, y, z) = (2x + 3y, y ? 2z).

Prove that f is a linear transformation. Describe Ker f and find a basis for
Ker f. Then, using the Rank and Nullity theorem, find the dimension of Im f?

I am having trouble with this, could someone explain to be the Ker f and how to go about this question thanks a lot!
2. (Original post by nkennedy)
Define a function f : R3 --> R2 by

f(x, y, z) = (2x + 3y, y ? 2z).

Prove that f is a linear transformation. Describe Ker f and find a basis for
Ker f. Then, using the Rank and Nullity theorem, find the dimension of Im f?

I am having trouble with this, could someone explain to be the Ker f and how to go about this question thanks a lot!
The kernel of f is the members of R^3 which when applied to the function f give the 0 vector in R^2. In this case then, f(x,y,z)=(0,0). So just solve this for x, y and z. (You will probably get multiple or infinite possible values of x, y and z and this is your kernel). The linear transformation part is easy - just show that f(ax,ay,az)=a*f(x,y,z) and f(x1+x2,y1+y2,z1+z2)=f(x1,y1,z1) +f(x2,y2,z2). As for the rank and nullity theorem, I don't know it too well, but it looks straightforward enough. Check wikipedia?
3. cheers for the help, very well explained (Y)

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