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Math 221: Matrix Algebra Midterm I - Sample Questions - January, 2013 1. TRUE or FALSE (a) A homogenous system with more variables than equations has a nonzero solution. (b) If x is a solution to the system of equations Ax = b for some vector b then 2x is also a solution to the same system. (c) If A has a zero column then the homogenous system Ax = 0 has a nonzero solution. (d) The only linear transformation which is both one-to-one and onto is the identity map. 2. Find the complete solution set to the system: 2x1 + 2x2 + x3 + x4 − x5 = 6 2x1 + 2x2 + x3 + 2x4 − 3x5 = 10 −2x1 − 2x2 + 2x3 − 3x4 + 6x5 = −12 3. Find the reduced row echelon matrix which is row equivalent to 3 0 2 1 3 3 3 1 2 3 8 14 10 3 5 12 4. Suppose 1 −3 0 2 0 0 1 3 0 A= 0 1 0 0 0 1 (a) Find all the solutions to the system of equations 2 Ax = −4 0 (b) Express the vector 2 −4 0 as a linear combination of columns of A. 5. In each of the following cases, determine whether the given vector is in the set spanned by the columns of the given matrix: 2 1 0 −1 (a) 1 with 0 1 −1 3 1 1 0 4 3 2 −4 (b) 0 with 1 −1 −3 −3 1 5 3 1 2 0 1 2 6. Determine if the set S = , , 3 is linearly independent in R3 and 2 3 4 explain why. 7. Determine if each of the following functions is a linear transformation. If it is the case find the matrix representing the transformation with respect to the standard bases. x1 (a) L : R3 → R1 , with L x2 = −x2 − x1 . x3 x1 x1 x2 2 2 (b) L : R → R , with L = . x2 x2 x1 (c) L : R3 → R with L x2 = 1. x3 x1 x2 − x1 2 2 (d) L : R → R with L = . x2 0 8. For the following linear transformations, find the standard matrix and also determine if they are one-to-one or onto. x1 (a) T : R3 → R with T x2 = x2 . x3 (b) T : R2 → R3 with T (e1 ) = e2 + e3 and T (e2 ) = −e1 + e2 . x x + x 1 1 2 (c) T : R2 → R2 with T = . x2 −x1 − x2 9. Suppose the following vectors in R3 are given 0 1 1 v 1 = 1 , v2 = 0 , v3 = 1 1 1 0 (a) Determine if the set S = {v1 , v2 , v3 } is linearly independent. (b) Determine if S spans R3 . (c) Express the vector 1 1 3 as a linear combination of elements of the vectors in S. 10. Suppose T : Rn → Rn is a linear transformation which is not onto. Answer the following questions and explain your answers. (a) What is the size of the standard matrix for T ? (b) How many pivots does the standard matrix of T has? (c) Can T be one-to-one?