{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

sample_tt1_2_ans

# sample_tt1_2_ans - Math 235 Sample Term Test 1 2 Answers...

This preview shows pages 1–2. Sign up to view the full content.

Math 235 Sample Term Test 1 - 2 Answers NOTE : - Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) Let A = 1 0 0 - 1 0 0 1 1 0 0 0 0 . Write a basis for the Row( A ), Col( A ) and Null( A ). Solution: A basis for Row( A ) is 1 0 0 - 1 , 0 0 1 1 . A basis for Null( A ) is 0 1 0 0 , 1 0 - 1 1 A basis for Col( A ) is 1 0 0 , 0 1 0 . b) Let B = { v 1 , . . . , v n } be orthonormal in an inner product space V and let v = a 1 v 1 + · · · + a n v n . Prove that a i = < v, v i > . Solution: Taking the inner product of both sides with v i to get < v, v i > = < a 1 v 1 + · · · + a n v n , v i > = a 1 < v 1 , v i > + · · · + a n < v n , v i > = a i since B is orthonormal. c) State the Rank-Nullity Theorem. Solution: Suppose that V is an n -dimensional vector space and that L : V W is a linear mapping into a vector space W . Then rank( L ) + Null( L ) = n. d) Find the rank and nullity of the linear mapping T : P 2 M (2 , 2) defined by T ( a + bx + cx 2 ) = c b 0 c . Solution: rank( T ) = 2, nullity( T ) = 1. 2. Let L : M (2 , 2) M (2 , 2) be given by L ( A ) = 1 2 3 4 A T . Find the matrix for L relative to the standard basis B of M (2 , 2) , where B = 1 0 0 0 , 0 1 0 0 , 0 0 1 0 , 0 0 0 1 .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

sample_tt1_2_ans - Math 235 Sample Term Test 1 2 Answers...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online