Practice Exercises PDF

Practice Exercises Exercise 1. (a) Construct a 3 × 3 matrix with rank 2 such that the column space is orthogonal to the null-space (b) Construct a 3 × 3 matrix with rank 1 such that the column space is orthogonal to the null-space (c) Can there be a 3 × 3 matrix such that the column spa...

Practice Exercises Exercise 1. (a) Construct a 3 × 3 matrix with rank 2 such that the column space is orthogonal to the null-space (b) Construct a 3 × 3 matrix with rank 1 such that the column space is orthogonal to the null-space (c) Can there be a 3 × 3 matrix such that the column space and null-space are the same? If yes, construct an example. If no, explain why. (d) Can there be a 4 × 4 matrix such that the column space and null-space are the same? If yes, construct an example. If no, explain why. (e) Construct a matrix whose column space has basis [1, 1, 0] and [1, 0, 1], but does not contain [1, 1, 1]. Exercise 2.   −1 0 1 A =  3 0 −3   1 0 −1 (a) Find a basis for row(A), col(A) and null(A) (b) Find an invertible matrix P and a diagonal matrix D such that A = P DP −1 " # a b Exercise 3. Let A =. c d 1. Prove that A is diagonalizable if (a − d)2 + 4bc > 0 and is not diagonalizable if (a − d)2 + 4bc < 0. 2. Find two examples to demonstrate that if (a − d)2 + 4bc = 0, then A may or may not be diagonalizable. Exercise 4. True or false (justify with a proof or a counter-example) (a) Let S be a subspace of Rn. Then the set of vectors of Rn not in S, i.e. Rn \ S is never a subspace of Rn. (b) For any n × n matrix A and non-zero scalar c ∈ Rn , col(A) = col(cA) and row(A) = row(cA) (c) For any n × n matrix A, col(A) = col(A − I) (d) A square matrix A is invertible if and only if 0 is not an eigenvalue of A Exercise 5. Verify if the following " # are sub-spaces. If they are, find a basis. x (a) The set of all vectors in R2 with xy ≥ 0 (i.e., the union of the first and third y quadrants). 1    a      2a − b      (b)   a, b, c ∈ R in R4 a−c           b+c   " # x (c) The set of all vectors in R2 with x ≥ y. y   a+4      b−4      (d)   a, b ∈ R in R4 −b + 4          a+4     x (e) The set of all vectors    in R3 with z = 5y. y  z x   " #  y  x y (f) The set of all vectors   in R4 such that the matrix is symmetric    z  z w w Exercise 6. 1 1 1 1         0 1 1 1 ⃗v1 =   ⃗v2 =   ⃗v3 =   ⃗v4 =           0 0 1 1 0 0 0 1 (a) Is {⃗v1 , ⃗v2 , ⃗v3 , ⃗v4 } a basis for R4. (b) Is {⃗v1 , ⃗v2 , ⃗v3 , ⃗v4 } orthogonal? (c) Let {v1 ,... , vn } be a basis for Rn. Prove that {v1 , v1 + v2 , v1 + v2 + v3 ,... , v1 + · · · + vn } is also a basis for Rn. (d) If {v1 ,... , vn } is orthogonal, is it true that {v1 , v1 + v2 , v1 + v2 + v3 ,... , v1 + · · · + vn } is also orthogonal Exercise 7. What is the determinant of n × n matrix A in the following situations: (a) If A is diagonalizable and has only one eigenvalue λ (b) A3 = I             1 1 −1 0 0 0 (c) A 1 = 0, A  1  = 1 and A 0 = 0             0 0 0 0 1 1 2