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Unformatted text preview: MAT067 University of California, Davis Winter 2007 Homework Set 4: Exercises on Linear Maps Directions : Please work on all of the following problems! Hand in the Calculational Prob- lems 1 and 2, and the Proof-Writing Problems 6 and 7 at the beginning of lecture on February 2, 2007. As usual, we are using F to denote either R or C . 1. Define the map T : R 2 R 2 by T ( x, y ) = ( x + y, x ). (a) Show that T is linear. (b) Show that T is surjective. (c) Find dim null T . (d) Find the matrix for T with respect to the canonical basis of R 2 . (e) Show that the map F : R 2 R 2 given by F ( x, y ) = ( x + y, x + 1) is not linear. 2. Consider the complex vector spaces C 2 and C 3 with their canonical bases. Let S : C 3 C 2 be defined by the matrix M ( S ) = A = parenleftbigg i 1 1 2 i 1 1 parenrightbigg . Find a basis for null S . 3. Give an example of a function f : R 2 R having the property that a R , v R , f ( av ) = af ( v ) but such that f is not a linear map.is not a linear map....
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