you need to prove that and , for starters.
f: R2 --> R2
X1 --> X1
X2 --> 1/3X2
There's only meant to be 1 arrow between the two matrices.
Prove that f is a linear map using additivity and homogeneity.
Determine the matrix which represents this linear map.
I have no idea what to do :s.
Any help would be greatly appreciated!
Thank you!
Ok, thanks. Any more suggestions after that?
I'm an exchange student and i cannot follow the lectures in the language set by the university, i'd appreciate if you could go through this example maybe so i could attempt other questions similar.
I know that to prove additivity, you need to show that for any real x_1, x_2, y_1, y_2 you have f{x_1\choose x_2} + f{y_1\choose y_2} = f{x_1+y_1\choose x_2+y_2}
and to prove homogeneity, you need to show that for any real x_1, x_2, alpha you have f{\alpha x_1\choose \alpha x_2} = \alpha f{x_1\choose x_2}, i need to evaluate the left and right hand sides of the equation and show that i get the same answer.
However, i am not sure how to evaluate the left and right hand sides of the equations? could you clarify this in detail or actually solve it please?
the idea of a linear map, is that it preserves vector sums, and scalar multiples. note that:
and are just notational differences, one can either represent a vector in horizontally or vertically, either way we need "two coordinates" (and this number-of-coordinates thing, is actually important, it's the main distinguishing feature of ).
to create a "scalar multiple" of , (no matter which way you write it), you multiply each coordinate by the scalar (which in this case means a real number, rather than a 2-vector).
so we should (if f is linear), get the same thing if we "do f first" and then multiply by a scalar, or if we multply by a scalar first, and then do f to the "scaled vector".
just as additivity means we can do f to each part of a sum separately, and then add, or we can add first, and then do f. some books phrase this as:
"f respects vector addition and scalar multiplication".
and are just vectors in , actually it doesn't matter how you call them, you just have to realise we're working in a 2-dimensional vector space so there are two components.
I think you can finish the proof now? (show us)
To determine the matrix which represents the linear map, have you any ideas about that? (think about bases of vector spaces).
we are not multiplying f and lambda. what we need to prove is:
.
f is a function that takes as its input 2 variables, and outputs two values. to prove the equality above, we need to calculate what f(___) is, using the definition of f.
so first we calculate:
then we use the definition of f, to calcuate the two values of f(λx):
now start on the other side with f(x) = f(x1,x2), calculate it, and then multiply each coordinate by λ, what do you get?