Math254Mid3Sols

# Math254Mid3Sols - MIDTERM 3 - SOLUTIONS MATH 254 - SUMMER...

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MIDTERM 3 - SOLUTIONS MATH 254 - SUMMER 2002 - KUNIYUKI CHAPTERS 6, 7 GRADED OUT OF 75 POINTS ¥ 2 = 150 POINTS TOTAL 1) The linear transformation TR R : 23 Æ is such that T 11 3 41 ,, , ()( ) =- and T 01 153 , () ( ) = . Find T 45 , . Hint: Remember the definition of a linear transformation. (5 points) Express (4,5) as a linear combination of the given basis vectors for R 2 , (1,1) and (0,1). Observe that (4,5) = (4,4) + (0,1) = 4(1,1) + (0,1). TT 411 41 1 0 1 43 4 1 1 53 12 16 4 1 5 3 11 21 7 , , , = + = + + + 2) The linear transformation R : 22 Æ is such that Tv v v v v v 12 1 2 1 2 3 ( ) + . Find the preimage of 17 5 , . (7 points) We want to solve the system (maybe by using Gauss-Jordan elimination) 21 7 235 vv -= += Ï Ì Ó 17 5 - È Î Í ˘ ˚ ˙ Subtract Row 1 from Row 2. RR R 2 1 +- Æ . 04 17 12 - - È Î Í ˘ ˚ ˙ Divide Row 2 through by 4. 17 3 - - È Î Í ˘ ˚ ˙ Add Row 2 to Row 1. RR R 1 .

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20 01 14 3 - È Î Í ˘ ˚ ˙ Divide Row 1 through by 2. 10 7 3 - È Î Í ˘ ˚ ˙ The preimage is 73 , - () {} . 3) The linear transformation TR R : 57 Æ is such that dim Ker T = 3 . (11 points total) a) What is the domain of T ? (2 points) R 5 b) What is nullity T ? (2 points) 3, since nullity( T ) = dim(Ker( T )). c) What is rank T ? (3 points) 2, since rank( T ) + nullity( T ) = dimension of domain = 5. d) True or False: Range T is a subspace of R 7 . Circle one: (2 points) True False Note that Range( T ) = Col( A ), where A is a 75 ¥ matrix.
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## This note was uploaded on 09/08/2011 for the course MATH 254 taught by Professor Howard during the Spring '09 term at Mesa CC.

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Math254Mid3Sols - MIDTERM 3 - SOLUTIONS MATH 254 - SUMMER...

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