Su SU19 M303 Practice Midterm 1. Multiple Choice. (a) Let A be any 5 × 4 matrix, fill in the following blanks with one of the three choices: (A) must (B) must not (C) may or may not i. A system Ax = b be consistent. ii. If a system Ax = b is consistent, it have a unique solution. iii. The column vectors of A be linearly dependent. iv. The column vectors of A span the entire R5 . v. The linear transformation x 7→ Ax be onto. vi. The linear transformation x 7→ Ax be one-to-one. (b) Let A be a m × n matrix. If Ax = b has a unique solution, choose ALL the correct statements from below. (A) (B) (C) (D) The row echlon form of A must have a pivot in each row. The row echlon form of A must have a pivot in each column. The vector b must be in the span of the column vectors of A. The vector b can be uniquely written as a linear combination of the column vectors of A. (c) If a n × n matrix A is not invertible, which of the following must be true? Choose ALL the correct statements. (A) (B) (C) (D) Ax = b is inconsistent for every b ∈ Rn . Ax = b either has no solution or infinitely many solutions. The column vectors of A are linearly independent. The determinant of A is zero. 1 2. Short answer. (a) Give an example of a 3 × 3 matrix A so that the solution set of Ax = 0 is a plane in R3 . (b) Give an example of a linear transformation T : R3 → R2 that is onto. x1 T x2 = x3 1 0 (c) Given vectors v1 = −1 and v2 = 1 . Find a vector v3 so that v1 , v2 , v3 0 −1 3 span the entire R . 1 3 2 5 2 3 6 , B = 5 2 1 3 (d) Let A = 4 1 2 0 4 1 2 × The dimension of the matrix AB is The (2, 3) entry of AB is . . (e) Let A, B, C, M be n × n matrices, solve M from the matrix equation CBM A = I. M= 2 a 1 0 1 1 0 , 1 0 , 1 , v is linearly 3. If v = b is a vector such that v ∈ span , and c 2 2 0 1 independent. What can you say about the values of a, b and c. 4. Find the matrix of the linear transformation T : R2 → R2 that rotates the plane counterclockwise by 3π/2. 5. Let A be a n × n invertible matrix. Prove that det(A−1 ) = 3 1 det(A) 3 −1 1 2 0 3 0 1 6. Let A = 5 1 0 2. 2 1 0 1 (a) Find the determinant of A. Based on your answer, is A invertible or not? (b) Based on your answer for part (a), does Ax = 0 have any nontrivial solution? 7. Let T : R2 → R2 be a linear transformation defined as T (x1 , x2 ) = (4x1 + x2 , 2x1 + 3x2 ). (a) If S is a region in R2 with area 2, what is the area of the image of S under T ? (b) Is T invertible? If so, find the standard matrix for T −1 . 4