oldprelim1 - A be a 5 × 5-matrix For(a(b and(c below say...

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Math 221 - Prelim 1- Feb 22, 2005 No notes. No calculators. No books. WORK + ANSWER = CREDIT 1. (24) (a) Solve the following system of equations: x + 2 y - 5 z = - 3 3 x + 6 y + 3 z = 9 - 2 x - 4 y + z = - 3 (b) Consider the following system of equations: x - 3 y = k - x + 2 y = 4 3 x + 2 y = 12 Find all values for k so that the above system of equations has a unique solution. 2. (20) Suppose A is a 5 × 3 matrix with rank 3. (a) What is the reduced row-echelon form of A ? (b) Suppse that b is a vector in R 5 . What can you say about the number of solutions of the system Ax = b ? Explain your answer. 3. (20) Let T be a linear transformation from R 2 to R 2 such that T - 1 1 = 2 1 and T 1 1 = 0 1 . (a) Find T 1 0 and T 0 1 . 1
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(b) What is the matrix A such that T ( x ) = A · x ? 4. (24) Find vectors that span the kernel of - 3 6 - 1 1 - 7 1 - 2 2 3 - 1 2 - 4 5 8 - 4 . 5. (24) Let
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Unformatted text preview: A be a 5 × 5-matrix. For (a), (b), and (c) below say whether it is true or false. You need not give any reasons. (a) If A is invertible, then im ( A ) = R 5 . (b) If A is invertible, then ker ( A ) = { ~ } . (c) If ker ( A ) = { ~ } , then im ( A ) is not R 5 . 6. (18) (a) Is the unit sphere U = x y z : x 2 + y 2 + z 2 = 1 a subspace of R 3 ? Explain. (b) Is W = ±² x y ³ : xy ≥ ´ a subspace of R 2 ? Explain. 7. (20) Fill the missing entries (the ? ’s) of the matrix A and its inverse A-1 A = 1 1 * * 2 3 * 3 * A-1 = * * 1-3 5 * 1-2 * . 2...
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