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exam2sol

# exam2sol - Math 51 Spring 2009 Exam II Solutions Problem...

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Math 51 — Spring 2009 — Exam II Solutions Problem 1. (10 pts.) Let L be a linear transformation from n to k . Each of the statements about L below is either always true (“T”), or always false (“F”), or sometimes true and sometimes false, depending on the situation (“MAYBE”). For each part, decide which, and justify your answer completely. Each part is 5 points: 2 for a correct answer, 3 more for correct justification. Note: if the answer is “MAYBE,” complete justification must include two examples — one where the statement holds true, and another where it is false. If the answer is “TRUE” or “FALSE,” a proper justification cannot be done by giving a single example — only by giving a proof. a) If vectors v 1 , v 2 , . . . , v m are linearly independent, then the vectors L ( v 1 ) , L ( v 2 ) , . . . , L ( v m ) are linearly independent as well. T F MAYBE As noted above, we must give two examples: one where this statement holds, and one where it does not. There are many possibilities for each. Suppose n = k and L = Id n , the identity transformation; then L ( v 1 ) = v 1 , L ( v 2 ) = v 2 , . . . , L ( v m ) = v m for any choice of vectors. So if we assume v 1 , v 2 , . . . , v m are linearly independent, then certainly L ( v 1 ) , L ( v 2 ) , . . . , L ( v m ) are independent as well. If we take L to be the zero transformation ( L ( v ) = 0 for all v ), then L ( v 1 ) = · · · = L ( v m ) = 0 , and so in this case L ( v 1 ) , . . . , L ( v m ) are linearly dependent. b) If vectors v 1 , v 2 , . . . , v m are linearly dependent, then the vectors L ( v 1 ) , L ( v 2 ) , . . . , L ( v m ) are linearly dependent as well. T F MAYBE Since v 1 , . . . , v m are linearly dependent, we may find scalars c 1 , . . . , c m , not all equal to 0, such that c 1 v 1 + · · · + c m v m = 0 . But then 0 = L ( 0 ) = c 1 L ( v 1 ) + · · · + c m L ( v m ), since L is a linear transformation, and so L ( v 1 ) , . . . , L ( v m ) are linearly dependent (because not all c i are equal to 0). 1

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Problem 2. (12 pts.) a) Complete the following sentence to make a true statement: A linear transformation L : n m is invertible if and only if (4 points) Choose your favorite one; they’re all equivalent (but be careful to note the dis- tinction between the function L and its matrix — these are different objects): . . . L is both one-to-one and onto. . . . n = m and L is onto. . . . n = m and L is one-to-one. . . . the matrix of L with respect to the standard basis (that is, the matrix A such that L ( x ) = A x for all x in n ) has reduced row echelon form equal to I n . . . . the matrix of L is square and has rank n . . . . the matrix of L is square and has linearly independent columns. . . . the matrix of L is square and has linearly independent rows. . . . the matrix of L is square and has nonzero determinant. and so on... For parts (b) and (c), suppose S is the linear transformation given by multiplication by the matrix A = 2 - 1 0 3 0 1 , and T is the linear transformation given by multiplication by the matrix B = 1 - 1 - 1 1 3 2 .
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