Assignment3Solutions(1) - MAT224 Assignment 3 Solutions 1...

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MAT224 Assignment 3 Solutions 1. Exercise #8 of 2.5. (You should use Proposition 2.5.13 for Part (b).)
Exercise. Suppose instead β is the basis { 1, x + 1, x 2 + x + 2, x 3 + x, x 4 } of P 4 ( R ) . Find [ Int D ] β β again. 2. (a) Suppose S : U V and T : V W are linear maps. Show that if T is injective, then ker ( S ) = ker ( TS ) . (b) Suppose S : U V and T : V W are linear maps. Show that if TS = 0 then Im ( S ) ker ( T ) . (By TS = 0 we mean that the composition TS : U W is the zero map, i.e. the map that sends every element of U to 1
2 0 W . Don’t be surprised if your argument is very short.)

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