Math 265 Linear Algebra  Sample Final  Fall 2003

1. It is given that A = [ \begin{matrix}1 & 0 & −2 & 1 & 3 \\ −1 & 1 & 5 & −1 & −3 \\ 0 & 2 & 6 & 0 & 1 \\ 1 & 1 & 1 & 1 & 4 \end{matrix} ], rref (A) = [ \begin{matrix}1 & 0 & −2 & 1 & 0 \\ 0 & 1 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \end{matrix} ].

(a) Find the rank of A.

(b) Find the nullity of A.

(c) Find a basis for the column space of A.

(d) Find a basis for the row space of A.

(e) Find a basis for the null space of A.

(f) Find a basis for the orthogonal complement of the row space of A.

2. Find an equation relating a, b and c so that the linear system

\{ \begin{matrix}2x & + & 2y & + & 3z & = a \\ 3x & − & y & + & 5z & = b \\ x & − & 3y & + & 2z & = c \end{matrix}

is consistent for any values a, b and c which satisfy that equation.

3. Determine the values of a so that the linear system

\{ \begin{matrix}x & + & y & + & z & = 2 \\ 2x & + & 3y & + & z & = 5 \\ 2x & + & 3y & + & (a^{2} − 3)z & = a + 1 \end{matrix}

has (a) no solution, (b) a unique solution, and (c) infinitely many solutions.

4. Find the value of y in the solution of the linear system

\{ \begin{matrix}2x & + & 123y & − & z & = 1 \\ x & + & 456y & + & z & = 0 \\ 2x & + & 789y & + & z & = 1 \end{matrix}

if you know that the deteminant of the matrix of the above system is equal to −420.

5. Let L : ℝ^{3} → ℝ^{3} be the linear transformation such that

L ([ \begin{matrix}1 \\ 0 \\ 0 \end{matrix} ]) = [ \begin{matrix}−3 \\ 1 \\ 2 \end{matrix} ] , L = ([ \begin{matrix}0 \\ 1 \\ 0 \end{matrix} ]) = [ \begin{matrix}2 \\ −2 \\ 1 \end{matrix} ] , L ([ \begin{matrix}0 \\ 0 \\ 1 \end{matrix} ]) = [ \begin{matrix}−1 \\ 2 \\ 4 \end{matrix} ] .

Find the standard matrix for L.

6. Let L : ℝ^{3} → ℝ^{3} be the linear transformation such that

L ([ \begin{matrix}−2 \\ 1 \\ −2 \end{matrix} ]) = [ \begin{matrix}−3 \\ 1 \\ 2 \end{matrix} ] , L = ([ \begin{matrix}3 \\ 2 \\ −1 \end{matrix} ]) = [ \begin{matrix}2 \\ −2 \\ 1 \end{matrix} ] , L ([ \begin{matrix}−1 \\ −1 \\ 1 \end{matrix} ]) = [ \begin{matrix}−1 \\ 2 \\ 4 \end{matrix} ] .

Find L ([ \begin{matrix}1 \\ 1 \\ −1 \end{matrix} ]).

7. Find the standard matrix of the linear transformation L defined by

L ([ \begin{matrix}x \\ y \\ z \end{matrix} ]) = [ \begin{matrix}2x − z \\ x + 2y + z \\ 3x − y \end{matrix} ]

8. If A is a 5 × 5 matrix and det A = 3 find det(A^{2} · (adjA)^{2}).

9. Find a basis for S = {t^{2} + 1, t^{2} + 2t, 3t^{2} + t − 1}. Does 6t^{2} − 1 belong to span(S)?

10. For what values of d are the vectors [1, 3, d], [1, 1, 0] and [0, 1, 1] linearly independent?

11. Compute adj(A) and A^{−1} for A = [ \begin{matrix}2 & 0 & 1 \\ −2 & 1 & 3 \\ 2 & −1 & 1 \end{matrix} ].

12. Find ALL diagonal matrices similar to A = [ \begin{matrix}1 & 1 \\ −5 & 7 \end{matrix} ].

13. Let us consider a triangle Δ(P, Q, R) in ℝ^{3} whose vertices are as follows: P = (−2\sqrt{3}, 1, −2), Q = (3\sqrt{3}, −3, 1), R = (−2\sqrt{3}, 7, 6)

(a) Find length of all sides of Δ(P, Q, R).

(b) Find all angles of Δ(P, Q, R).

14. Use the Gram-Schmidt process to construct an orthonormal basis for the subspace W

of ℝ^{4} spanned by

\{[ \begin{matrix}1 \\ 1 \\ 0 \\ 0 \end{matrix} ] , [ \begin{matrix}2 \\ −1 \\ 0 \\ 1 \end{matrix} ] , [ \begin{matrix}3 \\ −3 \\ 0 \\ −2 \end{matrix} ] , [ \begin{matrix}1 \\ −2 \\ 0 \\ −3 \end{matrix} ]\} .

15. Let W be the subspace of ℝ^{3} spanned by the vector w = [1 1 − 1].

(a) Find a basis for the orthogonal complement W^{⊥} of W.

(b) Find an orthonormal basis for W^{⊥}.

16. Let W be the subspace of ℝ^{4} spanned by the sset of vectors \{[ \begin{matrix}1 \\ 2 \\ 0 \\ −2 \end{matrix} ] , [ \begin{matrix}0 \\ 1 \\ 2 \\ 1 \end{matrix} ] , [ \begin{matrix}4 \\ −1 \\ 0 \\ 1 \end{matrix} ]\} ,

and v = [ \begin{matrix}1 \\ −1 \\ 0 \\ 4 \end{matrix} ]. (a) Find the projection of v onto W. (b) Find the distance from v

to W.

17. For each of the following sets of vectors, determine if it is a vector (sub)space:

(a) The set of all vectors in ℝ^{4} with the property

2x_{1} + x_{2} − 3x_{3} + x_{4} = 0;  Yes  No
(b) The set of all vectors in ℝ^{4} with the property x_{1}^{3} = x_{2}^{3}, x_{3} = x_{4};  Yes  No

(c) The set of all vectors in ℝ^{3}, which have the form (0, a − b + c, 3b + c)

where a, b and c are arbitrary real numbers;  Yes  No

(d) The set of all polynomials P in the space of all polynomials of degree

at most 7, with the property P (2) = 0;  Yes  No
(e) The set of all nonsingular matrices in the space of 3 × 3 matrices;  Yes  No

18. Determine if each of the following sets of vectors is linearly independent or linearly dependent:

(a) \{[ \begin{matrix}2 \\ 1 \\ 1 \\ 3 \end{matrix} ] , [ \begin{matrix}3 \\ 0 \\ 1 \\ 1 \end{matrix} ] , [ \begin{matrix}4 \\ 1 \\ 1 \\ 2 \end{matrix} ]\};  Independent  Dependent

(b) \{[ \begin{matrix}1 \\ 0 \\ 0 \end{matrix} ] , [ \begin{matrix}1 \\ 1 \\ 1 \end{matrix} ] , [ \begin{matrix}0 \\ 4 \\ 5 \end{matrix} ]\};  Independent  Dependent

(c) \{[ \begin{matrix}1 \\ 0 \\ 0 \end{matrix} ] , [ \begin{matrix}1 \\ 1 \\ 1 \end{matrix} ] , [ \begin{matrix}0 \\ 4 \\ 4 \end{matrix} ]\};  Independent  Dependent

(d) \{[ \begin{matrix}1 \\ 2 \\ 5 \end{matrix} ] , [ \begin{matrix}2 \\ 1 \\ 7 \end{matrix} ] , [ \begin{matrix}0 \\ 1 \\ 1 \end{matrix} ] , [ \begin{matrix}−3 \\ 2 \\ −1 \end{matrix} ]\};  Independent  Dependent

(e) \{[ \begin{matrix}1 \\ 2 \\ −1 \end{matrix} ] , [ \begin{matrix}1 \\ 0 \\ 1 \end{matrix} ]\}.  Independent  Dependent

19. We have a 3 × 3 matrix A = [ \begin{matrix}a & 1 & 2 \\ b & 3 & 4 \\ c & 5 & 6 \end{matrix} ] with det(A) = 3. Compute the determinant

of the following matrix

(a) [ \begin{matrix}a − 2 & 1 & 2 \\ b − 4 & 3 & 4 \\ c − 6 & 5 & 6 \end{matrix} ]

(b) [ \begin{matrix}7a & 7 & 14 \\ b & 3 & 4 \\ c & 5 & 6 \end{matrix} ]

(c) [ \begin{matrix}a & 1 & 1 \\ b & 3 & 3 \\ c & 5 & 5 \end{matrix} ]

(d) A^{T}

(e) (5A)^{−1}

20. Let A = [ \begin{matrix}3 & −5 \\ 1 & −3 \end{matrix} ].

(a) Find all eigenvalues of A.

(b) For each eigenvalue above, find an eigenvector of A associated to it.

(c) Find a diagonal matrix D and a nonsingular matrix P such that P^{−1}AP = D.

(d) Find A^{47}.

21. Let A = [ \begin{matrix}2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{matrix} ].

(a) Find all eigenvalues of A.

(b) For each eigenvalue as above, find an eigenvector of A associated to it.

(c) Find a diagonal matrix D and an orthogonal matrix P such that P^{T} AP = D.

22. Determine an invertible matrix Â and a vector b̂ such that the solution to ÂX̂ = b̂ is the least squares solution to AX = b, where

A = [ \begin{matrix}1 & 0 \\ 0 & 1 \\ 1 & 1 \\ 1 & 0 \\ 1 & 0 \end{matrix} ] and b = [ \begin{matrix}1 \\ 1 \\ 1 \\ 1 \\ 1 \end{matrix} ] .

23. Find the least squares fit line for the points

(1, 2), (2, 1), (3, 3), (4, 3).

24. Consider the homogeneous linear system of differential equations x′(t) = Ax, where

A = [ \begin{matrix}2 & −1 \\ 8 & 8 \end{matrix} ].

(a) Find all eigenvalues of A.

(b) Find the general solution to the linear system of differential equations.

(c) Solve the initial value problem for the given conditions x_{1}(0) = 3 and x_{2}(0) = −10.