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Additional problems for math 250 assignment #13, focusing on determining whether functions are one-to-one and onto, as well as showing that various vector spaces are isomorphic. Students are required to show that specific functions are linear transformations, one-to-one, and onto. The document also mentions that every finite-dimensional vector space is isomorphic to rn.
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Math 250 Additional Problems for Assignment #13 Due December 1, 2009
(a) T : R → R defined by T (x) = x^3.
(b) Let V = {all people who have ever lived} and W = {all women who have ever lived}. Define a function T : V → W by T (x) = mother of x. (Ignore Adam and Eve.)
(c) T : M 2 × 2 → R defined by T (A) = det(A).
(a) Let V =
x y 0
(^) : x, y are real numbers
. We will show that V ∼= R^2.
We begin by defining a function:
T : V → R^2
by
T
x y 0
x y
Now you must show that T is (1) a linear transformation, (2) one-to-one, and (3) onto.
(b) Show that M 2 × 2 ∼= R^4. Define a function
T : M 2 × 2 → R^4
(Look in your classnotes for help.) Then you must show that T is (1) a linear transformation, (2) one-to-one, and (3) onto.
(c) Show that P 3 ∼= R^4. Define a function
T : P 3 → R^4
(Look in your classnotes for help.) Then you must show that T is (1) a linear transformation, (2) one-to-one, and (3) onto.
(d) Your work in parts (b) and (c) show that
M 2 × 2 ∼= P 3 ∼= R^4
These problems just illustrate the (very big) fact we mentioned in class: If dim(V ) = n, then V ∼= Rn. So basically, every finite-dimensional vector space is just Rn!!!
