mat22a-practiceexam1

Mat22a-practiceexam1 - solution(s 5(a(8 points If T R 3 → R 3 is a linear transformation that satisﬁes T 1 1 1 = 1-2 and T 1 = 5 5 ﬁnd the

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Math 22A UC Davis, Winter 2011 Prof. Dan Romik Practice questions for the 1/26 midterm Notes: The questions below represent a rough approximation to the style and diﬃculty level of the midterm exam. The questions should be solvable in 45 minutes (the duration of the midterm). You should not use written notes or a calculator. 1. (25 points) Find the general form of the solution of the linear system x 3 - 2 x 4 = - 2 x 1 + 4 x 2 + x 3 + 7 x 4 = 1 2 x 1 + 8 x 2 + 3 x 3 + 12 x 4 = 0 x 1 + 4 x 2 - x 3 + 11 x 4 = 5 2. (15 points) For each of the following matrices, ﬁnd an augmented ma- trix in Reduced Row Echelon Form (RREF) that is row-equivalent to it: (a) 1 2 3 4 5 0 0 1 0 0 6 0 0 0 0 0 7 0 0 0 0 0 7 0 (b) 0 0 1 3 0 2 0 2 3 0 0 1 3. (15 points) In R 4 , let u = 1 - 1 1 - 1 , v = 2 2 0 0 . Compute the follow- ing quantities: (a) || u || (b) || v || (c) u · v (d) the angle between u and v (e) || u + v || 2 1

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4. (15 points) Solve the linear system described by the augmented matrix ± 1 3 5 2 k 9 ² depending on a parameter k , as a function of k . That is, specify the values of k for which a solution exists, and write a formula for the
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Unformatted text preview: solution(s). 5. (a) (8 points) If T : R 3 → R 3 is a linear transformation that satisﬁes T 1 1 1 = 1-2 and T 1 = 5 5 , ﬁnd the values T 3 3 3 = ? and T 3 3 4 = ? (b) (7 points) Let a transformation T : R 3 → R 3 be deﬁned by T x y z = -2 y + z x + y . Find a 3 × 3 matrix M such that T ( u ) = Mu for any vector u ∈ R 3 . 6. (15 points) For each of the following pairs of matrices M,N , compute the matrix products MN , or specify if the product does not make sense given the dimensions of M and N . (a) M = ± 1 0 0 2 ² , N = ± 1 3 5 7 9 11 ² (b) M = ± 1 1 x-1-1-1 ² , N = a b c 2 (c) M = 1 2 3 4 5 6 7 8 9 , N = 0 0 3 0 2 0 1 0 1 (d) M = ± x y ² , N = ( 20 10 ) (e) M = ± 1 3 5 7 9 11 ² , N = ± 1 3 5 7 9 11 ² 3...
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This note was uploaded on 11/13/2011 for the course MATH 22a taught by Professor Chuchel during the Winter '08 term at UC Davis.

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Mat22a-practiceexam1 - solution(s 5(a(8 points If T R 3 → R 3 is a linear transformation that satisﬁes T 1 1 1 = 1-2 and T 1 = 5 5 ﬁnd the

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