Group  ^{List No.}^{Last Name}^{Name}  ^{Student No.}^{Department}  ^{Section}^{Signature} ^{:}^{:}  ^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{Code}^{Acad. Year}^{Semester}^{Coordinator}^{Date}^{Time}^{Duration}  M E T U Department of Mathematics^{BASIC LINEAR ALGEBRA}^{Makeup Exam}^{Math 260}^{2007-2008}^{Fall}^{S.F, S.O, A.S.}^{January 22 2008}^{14:00}^{120 minutes}  ^{4 QUESTIONS ON 4 PAGES}^{TOTAL 100 POINTS}^{1}  ^{2}  ^{3}  ^{4}

Show your work ! Partial credits will not be given for correct answers if they are not justified.

Question 1 (6+6+6+6=24 points)

Let P_{2} denote the space of polynomials {ax^{2} + bx + c|a, b, c ∈ ℝ}.

a) Show that (f|g) = f(0)g(0) + f(1)g(1) is NOT an inner product in P_{2}

b) Show that (f|g) = f(0)g(0) + f(1)g(1) + f(−1)g(−1) is an inner product in P_{2}

c) Find an explicit relation between the coefficients a, b, c which characterizes the polynomials f(x) = ax^{2} + bx + c in the orthogonal complement {x + 1}^{⊥} = {f ∈ P_{2}|(f|x + 1) = 0}, for the inner product defined in b).

d) Determine a basis of the orthogonal complement {x + 1}^{⊥}.

Question 2 (6+8+6+6=26 points)

a) Show that L : ℝ^{2×2} → ℝ^{2}, L(A) = (trA, detA) is NOT a linear map. b)Assuming that L : ℝ^{2×2} → ℝ^{2} is a linear map such that L ( \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} ) = (1, 2), L ( \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} ) = (2, 3), L ( \begin{matrix}0 & 0 \\ 1 & 1 \end{matrix} ) = (1, −1), L ( \begin{matrix}0 & 1 \\ 0 & 1 \end{matrix} ) = (0, 1), determine L ( \begin{matrix}a & b \\ c & d \end{matrix} ).

c) Find a basis of the null-space (the kernel) N(L) = {v ∈ ℝ^{2×2}|L(v) = 0}.

d) Find the matrix of L with respect to the standard bases in ℝ^{2×2} and in ℝ^{2}.

Question 3 (6+8+8=22 points)

Let A be a 3×3 matrix for which the vectors ( \begin{matrix}1 \\ −1 \\ 0 \end{matrix} ), ( \begin{matrix}0 \\ 1 \\ −1 \end{matrix} ), ( \begin{matrix}0 \\ 0 \\ 1 \end{matrix} ), are eigenvectors with respective eigenvalues 2, 1, and −1.

a) Find a matrix P such that P^{−1}AP is diagonal.

b) Find A.

c) Find det(A^{−1}).

Question 4 (6+8+8+6=28 points)

Consider the space of skew-symmetric matrices V = \{( \begin{matrix}0 & x & y \\ −x & 0 & z \\ −y & −z & 0 \end{matrix} ) , x, y, z ∈ ℝ\}.

a) Write down an explicit formula for the inner product (A|B) = tr(AB^{T}) in terms of the matrix coefficients of

A = ( \begin{matrix}0 & a_{1} & a_{2} \\ −a_{1} & 0 & a_{3} \\ −a_{2} & −a_{3} & 0 \end{matrix} ) and B = ( \begin{matrix}0 & b_{1} & b_{2} \\ −b_{1} & 0 & b_{3} \\ −b_{2} & −b_{3} & 0 \end{matrix} ).

b) Show that matrices M_{1} = ( \begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ −1 & 0 & 0 \end{matrix} ), M_{2} = ( \begin{matrix}0 & 1 & 0 \\ −1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ), and M_{3} = ( \begin{matrix}0 & 1 & 1 \\ −1 & 0 & 1 \\ −1 & −1 & 0 \end{matrix} ), form a basis of

V and orthogonalize this basis.

c) Find the coordinates of A = ( \begin{matrix}0 & 3 & 4 \\ −3 & 0 & 5 \\ −5 & −4 & 0 \end{matrix} ), with respect to the basis \{M_{1}, M_{2}, M_{3}\}.

d) Find the projection p(A) of matrix A in c) to the subspace ⟨M_{1}, M_{2}⟩ ⊂ V spanned by M_{1} and M_{2}.