[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. A 3×3 real SYMMETRIC matrix A satisfies A[ \begin{matrix}1 \\ 1 \\ 1 \end{matrix} ] = [ \begin{matrix}2 \\ 2 \\ 2 \end{matrix} ], A[ \begin{matrix}1 \\ −2 \\ 1 \end{matrix} ] = [ \begin{matrix}−1 \\ 2 \\ −1 \end{matrix} ],

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

(a) What are the eigenvalues, the trace, and the determinant of A?

(b) Find A.

Q. 2. (a) Find the eigenvalues and the eigenvectors of A = [ \begin{matrix}2 & 1 & 0 \\ 0 & 2 & 0 \\ 1 & 0 & 1 \end{matrix} ].

(b) Is A diagonalizable? If yes, write a basis for ℝ^{3} in which the matrix of the linear transformation represented by A is diagonal. If no, explain why not.

(c) Use the CAYLEY-HAMILTON theorem to find A^{−1} .

Q. 3. Let f(x_{1}, x_{2}) = 19x_{1}^{2} − 6x_{1}x_{2} + 11x_{2}^{2} − 2x_{1} − 6x_{2} − 3 .

(a) Determine the type of the curve f(x_{1}, x_{2}) = 0 .

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

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

(d) Write the equation of the curve f(x_{1}, x_{2}) = 0 in the (y_{1}, y_{2}) -coordinates in standard form by COMPLETING THE SQUARE.

Q. 4. (a) Find all solutions to the system of linear equations AX = B, that is,

[ \begin{matrix}1 & −2 & −3 & 5 \\ 1 & −2 & −2 & −1 \\ −2 & 4 & 7 & −16 \end{matrix} ] [ \begin{matrix}x \\ y \\ z \\ w \end{matrix} ] = [ \begin{matrix}1 \\ 3 \\ 0 \end{matrix} ] .

(b) Considering A as the matrix of a linear transformation T : ℝ^{4} → ℝ^{3} and using part (a), find a basis for the kernel of T , find the nullity of T , find the rank of T , find a basis for the image of T , and determine whether or not T is one-to-one.

Name Last Name: Student Number:

Q. 5. Each part is a separate question independent of the other parts. Give explanations in the first three parts.

(a) Is the set S = { f : f is continuous on [0, 1] with ∫_{0}^{1} f(t) dt ≥ 0 } a vector space?

(b) Is the set of functions F = {e^{−x}, 0, e^{x}} on the real line linearly independent?

(c) Is there a linear transformation T : ℝ^{2} → ℝ^{2} simultaneously satisfying T (1, 2) = (3, 4) , T (5, 6) = (7, 8) , and T (6, 8) = (10, 14) ?

(d) Show that in a Euclidean (inner product) space if ⟨x+y, x−y⟩ = 0 , then ∥x∥ = ∥y∥.

(e) Show that in a vector space if the set A = {u, v, w} is linearly independent, then the set B = {u + v, v + w, w + u} is also linearly independent.

(f) Let E = {e_{1} = (1, 0), e_{2} = (0, 1)} be the standard basis and F = {f_{1} = (1, 2), f_{2} = (3, 4)} be another basis for ℝ^{2} . Find the transition matrix Q from E to F . If the coordinates of a vector v is (5, 6) with respect to E, find the coordinates of v with respect to F BY USING Q.