[figure: 1956; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; M E T U; Mathematics Department; Math 260 – Basic Linear Algebra; Final Exam; 2001–2002; Fall Semester; SF, HTK, EK, MP; Monday, 14.1.2002; 13:00–15:30; 2.5 hours; 5 questions on 6 pages.; Name :; Last Name :; Signature :; Student Number :; Section :; Total; Q.1; Q.2; Q.3; Q.4; Q.5; SHOW ALL YOUR WORK.; 100; 20; 20; 20; 20; 20]

Q. 1. The first quadrant V = { (x, y) ∈ ℝ^{2} : x > 0, y > 0 } in the plane is a real 2 -dimensional vector space under the addition ⊕ and scalar multiplication ⊙ defined by (x, y) ⊕ (z, w) = (xz, yw) and r ⊙ (x, y) = (x^{r}, y^{r}) .

(a) What is the zero element of V and what is the negative of (x, y) ? Justify your answer.

(b) Is P = { (x, y) ∈ V : y = 1 } a subspace of V ? What about Q = { (x, y) ∈ V : xy = 1 }? How about R = { (x, y) ∈ V : y = 2x }? Justify your answer.

(c) Show that B = {(1, e), (e, 1)} is a basis for V .

(d) Show that T : V → ℝ given by T (x, y) = 5 ln x is a linear transformation, where the target space ℝ is with standard addition, scalar multiplication, and basis. Then find the matrix of T in the given bases.

Q. 2. (a) Determine the values of a and b for which the system AX = B given by

x − 2y + az − w = 1
−x + y − z + w = −1
(a + 1)y − a^{2}z + aw = 0
(b + 1)y − abz − a^{2}w = b

has (i) no solution, (ii) unique solution, and (iii) infinitely many solutions. Write all solutions in the second and third cases.

(b) Considering A as the matrix of a linear transformation T : ℝ^{4} → ℝ^{4} and using part (a), find a basis for the kernel of T and a basis for the image of T .

Q. 3. (a) Find the eigenvectors and the corresponding eigenspaces of

A = ( \begin{matrix}3 & −1 & −1 & −1 \\ −1 & 3 & −1 & −1 \\ −1 & −1 & 3 & −1 \\ −1 & −1 & −1 & 3 \end{matrix} ) .

(b) Is A invertible? Why?

(c) Find a matrix C with the property C^{−1} = C^{t} such that D = C^{−1}AC is diagonal and find D.

Q. 4. Let f(x_{1}, x_{2}) = x_{1}^{2} + 4x_{1}x_{2} − 2x_{2}^{2} + 2x_{1} + 4x_{2} + 1 .

(a) Find an orthogonal coordinate-change matrix P that diagonalizes the quadratic part of f .

(b) Write down formulas that express the old coordinates x_{1} and x_{2} in terms of the new coordinates y_{1} and y_{2} .

(c) Write the equation f(x_{1}, x_{2}) = 0 in the (y_{1}, y_{2}) -coordinates in standard form by completing the square.

(d) Describe the set of points in the plane satisfying f(x_{1}, x_{2}) = 0 .

Q. 5. Each part is a separate question independent of the other parts.

(a) Give an equation for span{(−1, 1, 0), (−1, 0, 1)} in ℝ^{3} .

(b) If a linear transformation T : V → V satisfies T^{2} = 0 , show that Im T ⊂ Ker T .

(c) A linear transformation T : ℝ^{2} → ℝ^{2} takes the square S with vertices (0, 0) , (2, 0) , (0, 2) , and (2, 2) to the rhombus R with vertices (0, 0) , (3, 1) , (3, −1) , and (6, 0) . What are the possible values of the determinant of the matrix of T ? Explain.

(d) Express det(cof A) and cof(cof A) in terms of det A and A.

(e) A linear transformation T : V → V satisfies ∥T (x)∥ = ∥x∥ for all x ∈ V . What are the possible values of the eigenvalues of T ? Use real scalars.