Homework Chapter 5 UTK – M251 – Matrix Algebra Fall 2003, Jochen Denzler, MWF 1:25–2:15, Ayres 318

1. Check if the following set of vectors in ℝ^{4} is linearly independent. Show your work. S = {[0, 1, 2, 3]^{T} , [1, 2, 3, 4]^{T} , [0, 2, 0, 2]^{T} }

2. Explain in one sentence why it can be seen at a glance that the set in the previous example is not a basis of ℝ^{4}.

3. Is S = {[1, 1, 1]^{T} , [2, −2, 1]^{T} , [4, 0, 3]^{T} } (a) linearly independent?

(b) a spanning set for ℝ^{3}?

(c) a basis for ℝ^{3}?

Each answer requires an explanation.

4. Show that S = {[1, 1, 1]^{T} , [2, −2, 1]^{T} , [1, 0, 1]^{T} } is a basis of ℝ^{3}.

5. Show that the set S = {x^{2}+1, 2x, 2x^{2}−3} is a basis of the vector space P_{2} of polynomials of degree at most 2.

6. Is the set S = {1, sin x, cos x} linearly independent in the vector space C^{0}(ℝ) of continuous real functions? – Hint: You have to address the question: If k_{1}1+k_{2} sin x+k_{3} cos x = 0 (this 0 means the zero function), in other words, if k_{1}1 + k_{2} sin x + k_{3} cos x = 0 for every x, can we conclude that k_{1} = k_{2} = k_{3} = 0?

7. Is the set S = {1, sin^{2} x, cos^{2} x} linearly independent in the vector space C^{0}(ℝ) of continuous real functions?

8. Find the coordinate vector of the vector [1, 2, 3]^{T} with respect to the basis S = {[1, 1, 1]^{T} , [2, −2, 1]^{T} , [1, 0, 1]^{T} } of ℝ^{3} from pblm. 4.

9. Find the coordinate vector of the polynomial x^{2} + x + 1 with respect to the basis S = {x^{2} + 1, 2x, 2x^{2} − 3} from pblm. 5.

10. Show that if a, b, c are non-zero vectors in ℝ^{3} that are mutually orthogonal, then {a, b, c} is linearly independent.

11. According to Thm. 5.2.2 in the book, the set of solutions to a homogeneous linear system in n unknowns is a linear subspace of ℝ^{n}, called the solution space of the linear system.

Find a basis for the solution space of the homogeneous linear system, and determine the dimension of the solution space.

\begin{matrix}3x_{1} − x_{2} − x_{3} + 3x_{4} = 0 \\ 4x_{1} + x_{2} − x_{3} + 2x_{4} = 0 \end{matrix}

12. Same question for the system

3x_{1} − x_{2} − x_{3} + 3x_{4} = 0
4x_{1} + x_{2} − x_{3} + 2x_{4} = 0
x_{1} + x_{2} + x_{3} − 4x_{4} = 0

13. The real vector space C^{2}(ℝ) contains all twice differentiable real functions whose second derviative is still continuous. In M231 you have learned (or will learn) that a function y = f(x) satisfies the condition f′′(x) − 3f′(x) + 2f(x) = 0 for all x, if and only if f is of the form f(x) = c_{1}e^{x} + c_{2}e^{2x}. You may just take this fact for granted in this class, and now this should be an easy problem:

Give a basis for the vector space W consisting of those functions that satisfy f′′(x) − 3f′(x) + 2f(x) = 0 for all x; and determine the dimension of W.

14. What is the dimension of ℝ^{3×3} (also known under the name M_{33}, the vector space of all 3 × 3 matrices? Give two different bases for this vector space.

15. S_{0}^{3×3} is the set of all symmetric 3 × 3 matrices whose trace is 0. Make sure that you understand that S_{0}^{3×3} is a subspace of ℝ^{3×3}. What is the dimension of S_{0}^{3×3}? Give a basis. (If you have forgotten the definition of the trace, it’s early in the glossary, or use the index in the book.)

16. Draw into one single picture of the plane ℝ^{2} the following objects:

(a) The solution set of the homogeneous linear system [ \begin{matrix}2 & 1 \\ −4 & −2 \end{matrix} ] [ \begin{matrix}x \\ y \end{matrix} ] = [ \begin{matrix}0 \\ 0 \end{matrix} ]

(with each solution being represented as a point in ℝ^{2}).

(b) The solution set of the inhomogeneous linear system [ \begin{matrix}2 & 1 \\ −4 & −2 \end{matrix} ] [ \begin{matrix}x \\ y \end{matrix} ] = [ \begin{matrix}3 \\ −6 \end{matrix} ]

(with each solution being represented as a point in ℝ^{2}).

(c) For two particular solutions of the inhomogenous system from part (b), according to your free choice, highlight the point representing them in color, and also draw the respective vectors from the origin to these points in the same color.

I want a decent picture that you would not be ashamed to xerox for your class if you were the teacher.

17. Study sections 5.5 and 5.6 as needed; let

A = [ \begin{matrix}3 & 2 & 1 & −1 & 4 \\ 5 & 8 & 7 & 3 & 6 \\ −2 & 1 & 2 & 3 & −3 \end{matrix} ]

Invent one non-zero vector that is in the row space of A, but is not among the rows of A. Also determine the dimension of the row space of A by giving an explicit basis for it.

18. Same matrix A as before. Determine a basis for the column space of A. (There are many correct solutions of course.) Give the dimension of this column space.

19. Same matrix A as before. Is the vector [1, 1, 1]^{T} in the column space of the matrix A? Show your work. (No credit for merely saying yes/no.)

20. Same matrix A as before. Determine the null space of A.

21. Draw the vectors u = [2, 1]^{T} and v = [−1, 1]^{T} in the plane. (Again we need a decent figure that you would not be ashamed to xerox for your class if you were the teacher. A unit length of 1 or 1.5 inches is perfectly appropriate.) First rotate each of them by 60 degrees counterclockwise (geometrically, using ruler and protractor), then reflect the resulting vectors in the line x_{2} = −2x_{1}. You should have one line and six vectors in your figure now. In a different color, I want you to start over with u and v, but this time first reflect them in the line x_{2} = −2x_{1}, and then rotate the result counterclockwise by 60 degrees. That will add 4 more vectors in a different color to the figure.

22. Same u and v as before. Refer back to the figure from the previous problem.

Find the matrix Q that represents the rotation by 60 degrees counterclockwise, and the matrix S that represents the reflection in the line x_{2} = −2x_{1}. (You had one problem of this type in Chapter 4 already and may refer back for help.) Calculate both products QS and SQ. Label all vectors in the previous problem with u, v, Qu, Sv, QSu, etc., as appropriate. Calculate the vectors QSu and SQu algebraically and compare with the figure.

23. Calculate the determinant

| \begin{matrix}a & 2 & 3 & b \\ 1 & −1 & 2 & 5 \\ 2 & 2 & −1 & c \\ d & 2 & 3 & −2 \end{matrix} |

in a reasonably efficient way using at least one row or column operation and at least one cofactor expansion of some row or column.