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Spring ’07/MAT 335/Exam 1 Name: Show all your work.
- (6pts) For the matrices A, B and C find the following expressions, if they are defined: a) A^2 C b) BBT^ c) 2C − BA
A =
B =
[
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C =
[
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- (6pts) The matrix A is given below. a) Find the inverse of A. b) Use the inverse to effortlessly solve the system below.
A =
[
]
2 x 1 +4x 2 = 1 7 x 1 −x 2 = 3
- (4pts) Find the cosine of the angle between the vectors a = (1, − 1 , 3 , 4) and b = (0, 4 , 5 , 2).
- (9pts) A system of linear equations is given below. a) Use the Gauss-Jordan method (that is, transform the augmented matrix to reduced row- echelon form) in order to solve the system. b) Write the solution in vector form. c) Describe the set of points in R^4 that the solution set represents.
3 x 1 +x 2 +13x 4 = 11 −x 2 −x 3 − 6 x 4 = − 1 2 x 1 +2x 2 +x 3 +17x 4 = 9
- (10pts) Are the following statements true or false? Justify your answer by giving a logical argument or a counterexample. a) If u is orthogonal to v, then ‖u + v‖^2 = ‖u − v‖^2 b) If A is a 3 × 5 matrix with at least 2 non-zero entries, then the solution set of the linear system Ax = 0 always has at most 3 parameters. c) If A is an n × n matrix and A^17 = I, then A is invertible.
Bonus. (5pts) Use a linear system to show that the vector (− 2 , 25 , 11) is a linear combination of vectors (2, 1 , 3) and (3, − 5 , 1) and find the coefficients that realize this linear combination.
Spring ’07/MAT 335/Exam 2 Name: Show all your work.
- (5pts) Evaluate the determinant by any (efficient) method: ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 3 2 3 − 1 − 4 4 7 2 3 15 0 0 2 5 2 0 0 − 4 − 3 0 0 0 2 − 1 4
- (3pts) If det A = −5 and A is a 2 × 2 matrix, find:
det A−^1 =
det(3A) =
det A^4 =
- (6pts) Let Ax = b be a linear system whose solution is given below (A is a 2 × 4 matrix). a) Write any two solutions of the system. b) Write the general solution of the system Ax = 0. c) State the vectors that span the solution space of Ax = 0.
x 1 = 3 − 2 s +4t x 2 = 7 +3s x 3 = − 1 +8s − 7 t x 4 = − 5 s +t
- (4pts) Let T : R^2 → R^2 be the rotation about the origin by 120◦. a) Write the standard matrix of this transformation. b) Find T (1, 3).
- (7pts) Write the standard matrices for the following linear operators. a) T : R^2 → R^2 , T dilates by 4 in the x-direction, then reflects in the line y = x. b) T : R^3 → R^3 , T rotates about the positive z-axis by 90◦, then reflects in the xz-plane.
- (4pts) Show that the set of vectors of form (a, b, 3 a − 2 b) is a subspace of R^3.
- (9pts) Are the following statements true or false? Justify your answer by giving a logical argument or a counterexample. a) If det A = 0, then λ = 1 cannot be an eigenvalue of A. b) If A is orthogonal, then det A 6 = 0. c) If T : R^2 → R^2 is a linear operator, then T (x · y) = x · y for every x, y in R^2.
Bonus. (5pts) Show that ∣ ∣ ∣ ∣ ∣ ∣ 1 x x^2 1 y y^2 1 z z^2
= (y − x)(z − x)(z − y)
- (9pts) A matrix A is given below. a) Find a basis for the row space of A. b) Find a basis for the nullspace of A. c) Verify that row(A)=null(A)⊥^ by showing that every basis vector for row(A) is orthogonal to every basis vector for null(A).
A =
- (5pts) Let W be the subspace of R^3 spanned by vectors (2, 1 , 4) and (1, − 1 , 0). Find a basis for W ⊥.
- (6pts) Let A be a 3 × 7 matrix. Answer the following and justify your answers. a) What is the biggest rank(A) could be? b) What is the smallest nullity(A) could be? c) Give an example of a 3 × 7 matrix whose nullity is 5.
- (4pts) Are the following vectors a basis for the subspace of R^5 that they span?
v 1 = (∗, ∗, ∗, ∗, 1), v 2 = (∗, ∗, ∗, 1 , 0), v 3 = (∗, ∗, 1 , 0 , 0)
- (4pts) Complete the vector (0, − 1 , 1) to a basis of R^3. (That is, find additional vectors with which (0, − 1 , 1) makes a basis.)
Spring ’07/MAT 335/Final Exam Name: Show all your work.
- (3pts) For the matrices A, B and C find the following expressions, if they are defined: a) ABC b) BBT^ C
A =
[
]
B =
C =
[
]
- (8pts) A system of linear equations is given below. a) Use the Gauss-Jordan method (that is, transform the augmented matrix to reduced row- echelon form) in order to solve the system. b) Write the solution in vector form. c) Write the solution of the homogeneous system (numbers on the right replaced by 0’s). What is the basis of this subspace? What is the dimension?
x 1 −x 2 +2x 3 −x 4 = − 1 2 x 1 +x 2 − 2 x 3 − 2 x 4 = − 2 −x 1 +2x 2 − 4 x 3 +x 4 = 1 3 x 1 − 3 x 4 = − 3
- (4pts) Evaluate the determinant by any (efficient) method: ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ 2 1 0 0 1 2 1 0 0 1 2 1 0 0 1 2
- (6pts) The matrix A is given below. a) Find A−^1. b) Use the result of a) to easily solve the system Ax = b, where b = (− 1 , 2 , 4).
- (4pts) Find the matrix of the linear operator T : R^3 → R^3 that is the composition of a rotation by 60◦^ about the positive x-axis, followed by a projection to the xz-plane.
- (5pts) Let T be the rotation about the origin in R^2 by 30◦. Find the vector that T sends to the vector (− 5 , 3).
- (8pts) Let A be a 5 × 3 matrix. Answer the following and justify your answers. a) What is the biggest rank(A) could be? b) What is the smallest nullity(A) could be? c) If TA is the linear transformation corresponding to A, is TA ever onto? Is it ever one-to- one?
- (3pts) Let E 1 be the matrix obtained from I 2 by adding 3 times row 1 to row 2 and let E 2 be be the matrix obtained from I 2 by swapping the two rows. Find the matrix below.
E 2 E 1
[
]
- (4pts) Let W be the subspace of R^4 spanned by vectors (1, 2 , 1 , 4) and (3, 1 , − 1 , 0). Find a basis for W ⊥.
