Test 2 Math 2407Spring 2011

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You may have one sheet of scratch paper, but the work must be written on this test. You may use acalculator (but no cell-phone or computer), but it isn't necessary . You may not use any notes norbooks. Remember that you must shield your test from anyone sitting around you (if you get calledon during the test to move, please do so).

1. a. (5 points) Give an example of a 4x3 matrix whose columns form a linearly independent set. Explainwhy your example satisfies the request or why no such matrix is possible.

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b. (5 points) Give an example ofa 3x4 matrix whose columns form a linearly independent set.Explain why your example satisfies the request or why no such matrix is possible.

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2. (8 points)(a) Give an example of a transformation! R2 ~ R2 that reflects vectors through the XI axis.

H!I)" [~_~}~ / +~(b) Give a geometric description of the matrix transformation! R 2

[-3 0]J(u) = A u when A = 0 3

~ R 2 defined by

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3. (14 points) Suppose that.Z' is a linear transformation from 2-space to 3-space, which satisfiesT(1,2) = (1, 3, 5) and T(2, 5). = (-I, I, I).

(a) Write vector (1,0) as linear combination of the vectors (1,2) and (2,5)

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(b) Write vector (0,1) as linear combination of the vectors (1,2) and (2,5)

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(c) Use T(1,2) = (1, 3, 5), T(2,5) = (-I, I, I), the results of parts (a) and (b), and the fact thatT(x,u+x2v) = xi T(u) + x2T(v) to evaluate T(1,O) and T(O,I).

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-r(t.}2.]:: T (-2~+~') t: ~2.T(~) r T(~) ~ -2..(I,S.S).I- (.-1;'I')

0:. (2""1-/'0') + (-I) I) \) z (-SJ -S,-&t'),

(d) Write the matrix representation for T.

4. (6 points) Let A, Band C be matrices. Mark each True or False

T A linear transformation is a function

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__ T_' _ If A is a 5 x 5 matrix and T: x H Ax is a linear transformation, then T(-Ix) = -IT(x).

5. (7 points) Let A, Band C be matrices. Mark each True or False

l=' ~atrix multiplication is commutative for n x n matrices

~the dimensions of A, Band C are compatible, then matrix multiplication is distributesover addition: A(B + C) = AB + AC (6vt- (B t- C)A 1- A B t- /4-c.')

.,- vMatrix transpose distributes over addition: (A + Bl = AT + BT

E ~ A and B are invertible, then matrix inversion distributes over addition:(A + B)-I = A-I + B-1 +1

~f A and B are invertible, then matrix inversion distributes over multiplication:(ABrl = A-I B-1

-, ~ A is invertible, then matrix inversion is satisfies: (A-Irl = A

T ~atriX inversion issatisfies: (A' r' ~(A -i )'

6. (10 points) Write five statements from the Invertible Matrix Theorem that are each equivalent tothe statement that an n x n matrix A is invertible. Use the following concepts once in each of yourstatements: (i) linear independence, (ii) pivots, (iii) span, (iv) one-to one, and (v) the equation Ax= O.

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7. (8 points) Give examples of three 2 x 2 matrices, A, Band C that demonstrate thatA(B + C) * (B + C)A. (verify that your examples demonstrate the request)

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8. (8 points) Solve the equation A (XB-1 + C) = D for X, assuming that A, B, C and D are invertiblematrices. (Beware of the distribution properties for matrices)

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=-) X~-I :: A-I D ... c:.') ~ \'?>- \ ~ ~ LA-\ D ~c')B

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9. (10 points) Compute the inverse of the matrix or show that the inverse doesn't exist

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5 -2 -3

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10. a. (5 points) Let A be an m x n matrix, and B = lbl

The number of entries in vector bp is _1t ~

The number of columns of matrix B is P

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b , ... b p J is an n xp matrix.

By definition AB = [ It 1;./ A a2 / t5b. (6 points) Suppose that A and Bare 5x5matrices, all of the entries inA and B are nonzero, but

the third column of the product AB is entirely zero. Could the matrix A be invertible? Justifyyour answer. [Hint: Think about the definition of the product AB of two matrices.]

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1L (8 points) Write a system of equation:that determines the loop currentsfor the following circuit network.

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1Co -r; - 2. 5 Iz. - 5DT3> -:; 10.,

-2$r, + 5G,IZ - X3 z - JD

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30 Q

55 Q