A2 - L : V W be a linear mapping. Prove that if { L ( ~v 1...

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Math 235 Assignment 2 Due: Wednesday, May 18th 1. Consider the projection proj (1 , - 2) : R 2 R 2 onto the line ~x = t ± 1 - 2 ² , t R . Determine a geometrically natural basis B and determine the matrix of the transformation with respect to B . 2. Find a basis for the range and kernel of the following linear mappings and verify the Rank-Nullity Theorem. (a) L : R 3 R 2 defined by L ( x 1 ,x 2 ,x 3 ) = ( x 1 + x 2 ,x 1 + x 2 + x 3 ). (b) L : R 3 P 1 defined by L a b c = ( a + b ) + ( a + b + c ) x . 3. Invent a linear mapping L : R 3 P 2 that satisfies L 1 0 0 = x 2 , L 0 1 0 = 2 x , L 0 0 1 = 1 + x + x 2 . 4. Let L : P 2 P 2 defined by L ( a + bx + cx 2 ) = a + ( b + c ) x 2 and let B = { 1 + x 2 , - 1 + x, 1 - x + x 2 } . Find [ L ] B . 5. Let L : R 2 P 2 defined by L ( a 1 ,a 2 ) = ( a 1 + a 2 ) + a 1 x 2 , B = ³± 1 - 1 ² , ± 1 2 ²´ , and C = { 1 + x 2 , 1 + x, - 1 - x + x 2 } . Find C [ L ] B . 6. (a) Let V and W be vector spaces and
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Unformatted text preview: L : V W be a linear mapping. Prove that if { L ( ~v 1 ) ,...,L ( ~v k ) } is a linearly independent set in W , then { ~v 1 ,...,~v k } is a linearly independent set in V . (b) Give an example of a linear mapping L : V W where { ~v 1 ,...,~v k } is linearly inde-pendent in V , but { L ( ~v 1 ) ,...,L ( ~v k ) } is linearly dependent in W . 7. Let U , V be nite dimensional vector spaces and let L : V U be a linear mapping and M : U U be a linear operator such that Ker( M ) = { ~ } . Prove that rank( M L ) = rank( L )....
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