Show that the mapping f: R^{2}->R^{2} given by f(x,y)=(x+y,x-y) is linear. For each subspace X of R^{2 }describe f^{-->}(X) and f^{<--}(X).
My question is whether f^{-->}(X) means that the image of the x-axis is the line y=x and the image of the y-axis is the line y=-x, so how am I describe f^{<--}(X)?
Thanks in advance.
The image of the x-axis (points of the form (x, 0)) is the set (x+ 0, x- 0)= (x, x) so y= x, and the image of the y-axis (points of the form (0, y)) is (0+ y, 0- y)= (y, -y) so y= -x.
The inverse image of a set X is the set of all points (x, y) such that f(x,y) is in X. If f(x,y)= (x+ y, x- y)= (u, v) then x+ y= u, x- y= v. Adding the two equations 2x= u+ v so x= (u+ v)/2. Subtracting x- y= v from x+ y= u, 2y= u- v so y= (u- v)/2. That is, which is the same as .
since we are dealing with R^{2}, which is finite-dimensional, we can use matrices. of course, to do that, we need to pick a basis, so we can use that basis to turn f into a matrix.
but the standard basis B = {(1,0),(0,1)} will do just fine.
note that f(1,0) = (1,1), and f(0,1) = (1,-1), so relative to the standard basis f has the matrix:
.
this has determinant -2 ≠ 0, so it has an inverse matrix M^{-1}, which corresponds to f^{-1} (relative to the same (standard) basis).
the inverse is found to be:
.
thus f^{-1}(x,y) = (1/2)(x+y,x-y), as the previous posts have already indicated.
R^{2} has a particularly simple subspace structure:
if U is a subspace of R^{2}, there are 3 possibilities:
U = R^{2} (the only 2-dimensional subspace)
U = {a(x_{0},y_{0}): a in R}, the set of all scalar multiples of a particular non-zero vector (x_{0},y_{0}) of R^{2} (1-dimensional subspaces)
U = {(0,0)} (the only trivial subspace).
since f is bijective, if U = R^{2}, then f(U) = f(R^{2}) = R^{2}, and f^{-1}(U) = f^{-1}(R^{2}) = R^{2}.
similarly if U = {(0,0)}, f(U) = {(0,0)} and f^{-1}(U) = {(0,0)}.
so the only "interesting" case is where U = {a(x_{0},y_{0}): a in R}.
by the linearity of f (or by the linearity of the matrix for f, which is the same thing), f(U) = {a(x_{0}+y_{0},x_{0}-y_{0}): a in R}.
for example if (x_{0},y_{0}) = (3,4), f(U) has basis {(7,-1)}.
interestingly enough, f^{-1} = (1/2)f, so f^{-1}(U) = f(U) (since for any non-zero vector v, {v} and {(1/2)v} generate the same subspace).
*********
while the x-axis and the y-axis are indeed each subspaces of R^{2} (as is any line through the origin), they are by no means the ONLY subspaces of R^{2}.
Thank you very much...your answers are explained clearly and help to put in order the information I have...
Just a point I wish to make sense out of it...According to the book (whence this question) the answer involves "the image of the line y=mx where m ≠ 1 is the line y=x(1+m)/(1-m)", how does it fit?
claim: the subset U = {(x,y) in R^{2}: y = mx} is a subspace of R^{2} of dimension 1.
proof:
we must show 3 things:
a) if (u,v) and (u',v') are in U, so is (u,v) + (u',v').
b) if c is any real number, and (u,v) is in U, so is c(u,v).
c) (0,0) is in U.
we start with (a). if (u,v) and (u',v') are in U, then v = mu, and v' = mu'.
thus (u,v) + (u'v') = (u,mu) + (u',mu') = (u+u',mu+mu'), and mu+mu' = m(u+u'), so (u,v) + (u'v') is in U.
now (b): if (u,v) is in U, then v = mu. so c(u,v) = c(u,mu) = (cu,c(mu)). since c(mu) = (cm)u = (mc)u = m(cu), we see c(u,v) is also in U.
and finally, (c): (0,0) = (0,m0), so (0,0) is in U.
this shows U is indeed a subspace of R^{2}. to prove it has dimension 1, we need to find a basis.
i claim {(1,m)} is a basis for U. clearly this is linearly independent, since (1,m) is a non-zero vector. so all we have to do is show {(1,m)} spans U.
again, suppose that (x,y) is any point of U. then, by the definition of U, y = mx, so (x,y) = (x,mx). thus (x,y) = (x,mx) = x(m,1), so {(m,1)} spans U.
claim 2: if U is a 1-dimensional subspace of R^{2}, then either U is the line y = mx, for some real number m, or U is the y-axis.
suppose U is not the y-axis. since U is one dimensional, some non-zero vector (a,b) is a basis for U. suppose that a ≠ 0.
let m = b/a. then for any vector (x,y) in U, (x,y) = c(a,b) = (ca,cb). thus y = cb = c(b/a)a = (b/a)(ca) = m(ca) = mx, that is: U is the line y = mx.
on the other hand, suppose our basis for U is (0,b), where b ≠ 0. then U = {(x,y) in R^{2}: (x,y) = c(0,b), for some c in R}.
but c(0,b) = (0,cb), so every point of U lies on the y-axis, and every point of the y-axis is in U, so U = the y-axis.
***********
so yes, the 1-dimensional subspaces of R^{2} are "lines through the origin" (perhaps explaining why its called "linear" algebra: bases consisting of a single vector are LINES, lines are the "building blocks" by which we make "bigger spaces" (planes, and 3-spaces, etc.)).
***********
now a linear map, takes lines (through the origin) to "some other lines" (also through the origin).
consider, what happens when we take f and apply it to the vector (x,mx) (that is: a point lying on the line y = mx).
we get f(x,mx) = (x + mx,x - mx).
which line is this?
we can use the "two-point" formula for a line:
which two points shall we use?
how about (x_{1},y_{1}) = f(0,0) = f(0,m0) = (0+m0,0-m0) = (0,0), and:
(x_{2},y_{2}) = f(1,m) = (1+m,1-m)
then our line formula becomes:
which simplifies to y = [(1-m)/(1+m)]x (perhaps there is a typo in your book?).
the "bad value" for m would be m = -1:
in this case, f maps (x,-x) to (0,2x), which lies on the y-axis (a line of "infinite slope", that is: vertical).