PracticeProblems2Fall2012.pdf

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Department
Mathematics
Course
MATH 1502
Professor
Blekherman
Semester
Fall

Description
Practice Problem for Midterm #2, Math 1502 1. Determine whether the given vectors are linearly independent or dependent. If the vectors are linearly dependent, express 0 as a nontrivial linear combination of the vectors: 0 1 0 1 0 1 1 2 -3 a)@ -2A ;@ -5A ; @ 2 A 3 8 -1 0 1 0 1 0 1 1 1 0 b)@ 1 A; @ 0 A ;@ 1 A 1 1 1 ▯ ▯ ▯ ▯ ▯ ▯ c) 1 ; 1 ; -3 -2 -5 2 0 1 0 1 2 -2 B C B C d)B -2C ;B 2 C @ 3 A @ -3A 4 4 2. For every system of equations given below perform the following steps: (a) Write the augmented matrix of the system. (b) Reduce the system to reduced echelon form. (c) Find the solutions of the system. (d) Express the solution in vector form. a) ▯2x2+ x3▯ x4= 3 x + x ▯ 2x + 3x = 2 1 2 3 4 ▯x1+ 3x3= ▯1 b) x1+ x2+ 4x3+ 3x4= 0 2x + 4x + 9x ▯ x = 0 1 2 3 4 c) x1+ x2+ x 3 5 ▯x 1 2x 2 2x 3 1 3x1▯ x 2 5x 3 3 c) x1+ x2= 0 ▯4x + 3x = 1 1 2 ▯3x 1 4x 2 ▯1 1 3. Find the matrix of a linear transformation T given the information below: ▯ ▯ ▯ ▯ ▯ ▯ ▯ ▯ (a) T 1 = 3 ; T 4 = 1 0 4 4 -3 0 1 0 1 0 1 0 1 0 1 0 1 0 2 0 1 1 2 @ 0 A @ 3A @ 1A @ -3A @ 1A @ 2 A (b) T = ; T = ; T = 1 -2 1 1 1 1 4. ▯ ▯ ▯ ▯ ▯ ▯ 1 1 2 (a) Is-3 in the span -1 and -3 ? If yes, ▯nd a speci▯c linear combination. 0 1 0 1 0 1 0 1 1 1 0 2 (b) Is -3A in the span of1A , -2A [email protected] 0 A? If yes, ▯nd a speci▯c linear -2 2 1 1 combination. 0 1 0 1 0 1 1 2 3 (c) Is 1A in the span of2A [email protected] -1A ? If yes, ▯nd a speci▯c linear combination. -2 3 1 5. For a linear transformations T ▯nd the following: 0 1 0 1 0 1 0 1 0 1 1 2 0 0 1 @ A @ A @ A @ A @ A (a) Given that T = 1 and T 1 = 0 ▯nd T -1 . 1 -1 1 2 0 0 2 1 0 1 1 0 6 1 (b) Given that T3A = @ -5A ▯nd [email protected] -9A. 1 6 3 0 1 0 (c) With no information, ▯nd.T 0 6. At a store a notepad costs $2 and a notebook costs $3. In a day 110 items were sold and $264 were spent. Find how many notepads were sold. 7. True or False. No partial credit. (a) The span of the columns of a matrix A is equal to the image of the linear transformation T given by T(x) = Ax. (b) Any 3 v
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