Linear Mappings: Solving for Dimension of Nullspace

[DylanB]

Homework Statement

http://img526.imageshack.us/img526/743/93134049.png

Homework Equations

Just the standard linear mapping properties and theorums.

The Attempt at a Solution

I have already solved part A by considering the transformation A: Rn -> R | A(x) = a.x where x is a vector in Rn, and finding the dimension of the nullspace.

I am stuck on where to begin the proof for part b, once I have b I don't think I will have a problem generalizing it for part c.

 

[ystael]

Your solution to part (a) already contains the essence of a solution to part (b); just think about the dimension of the null space of a different linear map [tex]B[/tex], one that uses the existence of [tex]\vec{a}[/tex] and [tex]\vec{b}[/tex] both.

(If you need a further hint: given that you are supposed to find that [tex]\dim(S \cap T) = n - 2[/tex] in case [tex]\vec{a}[/tex] and [tex]\vec{b}[/tex] are linearly independent, what do you think the dimension of the range space of [tex]B[/tex] ought to be?)

 

[DylanB]

ah, thank you, very good. I have a transformation B: Rn -> R2 that works nicely, and the generalization follows quite easily.