Linear Independence and Linear Functions

[catscradle]

I need some help with examples. Especially number 2.

1) Name a subset which is closed under vector addition and additive inverses but is not a subspace of R squared.

I think I got this one. {(x,y) st x,y are elements of integers} because this isn't closed under scalar multiplication

2) Give an example of a function f: R squared --> R that satisfies f(av) = af(v) for all a in the element of R and all v in the element of R squared, but f is not linear.

I don't really know what a linear function is. I tried f=0V, but my teacher said that's wrong. Any advice?

3) Find a set of 4 linearly independent vectors in the vector space L(R squared, R squared)

I think {3x+1, 4x, 8x^2 - x, 12x^3 - x^2} is an answer.

 

[HallsofIvy]

I need some help with examples. Especially number 2.

1) Name a subset which is closed under vector addition and additive inverses but is not a subspace of R squared.

I think I got this one. {(x,y) st x,y are elements of integers} because this isn't closed under scalar multiplication

Yes, that's correct. Very good!

2) Give an example of a function f: R squared --> R that satisfies f(av) = af(v) for all a in the element of R and all v in the element of R squared, but f is not linear.

I don't really know what a linear function is. I tried f=0V, but my teacher said that's wrong. Any advice?

A linear function (and if you were given this problem you were certainly expected to know what it is- perhaps you know them as "linear operators" or "linear transformations") is a function f defined on a vector space such that f(u+ v)= f(u)+ f(v) and f(av)= af(v) where u and v are vectors and a is a scalar. Since those are the operations on a vector space, linear functions "play nicely" with vectors and are the most important functions on vector spaces. f= 0V (f(v)= 0 for all vectors, v) is linear- that's why you teacher said that is wrong. Since f(av)= af(v) is one of the requirements for a linear function, it must be the other, f(u+ v)= f(u)+ f(v), that is violated. I'm not at all sure I can think of any good "hint" here. I'll just say, consider the function f((x,y)= ^3\sqrt{x^2y}.

3) Find a set of 4 linearly independent vectors in the vector space L(R squared, R squared)

I think {3x+1, 4x, 8x^2 - x, 12x^3 - x^2} is an answer.[/QUOTE]
What do you think L(R²,R²) means? It is the set of Linear functions from R² to R². (Which makes me very concerned about your statement "I don't really know what a linear function is"! Obviously your teacher thinks you should!)

Your examples are functions from R to R, not R² to R² and three of the four are not "linear". You need functions that take a vector <x,y> to another vector, something line f(<x,y>)= <f_x(x,y), f_y(x,y)> and, again, the functions must be linear. Are you at all familiar with representing linear functions as matrices? L(R², R²) can be represented as the set of all 2 by 2 matrices.