course_examples_299_wk10_solution - MTH299 Examples Week 10...

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MTH299 - Examples Week 10 Example 10.1. Is the set A = { ( x, y ) x, y R } a vector space over R under the following operations? (a) ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( x 1 + x 2 , y 1 + y 2 ) and r · ( x 1 , y 1 ) = ( rx 1 , y 1 ). (b) ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( x 1 + x 2 , y 1 + y 2 ) and r · ( x 1 , y 1 ) = ( rx 1 , 0). Here ( x 1 , x 2 ) , ( y 1 , y 2 ) A and r R .
Example 10.2. Define the set V = { ( x 1 , x 2 ) R 2 : x 2 2 x 1 + 1 } . Sketch a picture of the set V inside of the plane, R 2 . Is V a vector space over R ? Justify your answer!
MSU 1
MTH299 - Examples Example 10.3. Define V = { p P 2 : x R , p 0 (1) = 0 } . Is V a vector space over R ?

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