Group  ^{List No.}^{Last Name}^{Name}  ^{Student No.}^{Department}  ^{Section}^{Signature} ^{:}^{:}  ^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{Code}^{Acad. Year}^{Semester}^{Instructor}^{Date}^{Time}^{Duration}  M E T U Department of Mathematics^{Basic Linear Algebra}^{Final Exam}^{Math 260}^{1999-2000}^{Spring}^{5.1.1999}^{9:30}^{minutes}  ^{4 QUESTIONS ON 4 PAGES}^{TOTAL 100 POINTS}^{1}  ^{2}  ^{3}  ^{4}  ^{5}

\underline{(30 pts.) Q.1.} Let u = (1, −1, 1) ∈ ℝ^{3} and V = \{(a_{1}, a_{2}, a_{3}) | a_{1}+2a_{2}−a_{3} = 0, a_{2}−a_{3} = 0\}

Find the orthogonal projection of u on V. What is the shortest distance from u to V?

\underline{(40 pts.) Q.2.} Let A = ( \begin{matrix}−2 & −3 & −3 \\ −1 & 0 & −1 \\ 5 & 5 & 6 \end{matrix} )

(i)Find the eigenvalues of A. (ii) Find a basis for each one of the eigenspaces of A. (iii) Is A diagonalizable? If it is find an invertible matrix P diagonal matrix D such that P^{−1}AP = D . Otherwise explain why A is not diagonalisable.

\underline{(50 pts.) Q.3.} Let B = \{[ \begin{matrix}1 & 0 \\ 0 & 0 \end{matrix} ] , [ \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ] , [ \begin{matrix}0 & 0 \\ 0 & 1 \end{matrix} ]\} ,

V = spanB = \{[ \begin{matrix}a & b \\ 0 & c \end{matrix} ] | a, b, c ∈ ℝ\}

Let L : V → ℝ^{⊯} be a linear transformation whose matrix relative to the pair

B, E = \{(1, 0, 0), (0, 1, 0), (0, 0, 1)\} is A = [ \begin{matrix}2 & −1 & −1 \\ −1 & 2 & −1 \\ −1 & −1 & 2 \end{matrix} ]

a) Compute L ( \begin{matrix}a & b \\ o & c \end{matrix} )

b) Find a basis for S = \{( \begin{matrix}a & b \\ 0 & c \end{matrix} ) | L ( \begin{matrix}a & b \\ 0 & c \end{matrix} ) = 0\} \underline{(30 pts.) Q.4.} For the subsets of ℝ^{⋭}, defined below, determine whether they are subspaces or not. Show your reasoning

a) W = {(x, y, z, t) | x, z, t ∈ ℝ, ↷ ∈ ℤ is the set of integers.}

b) W = {(a, b, c, d) | a^{2} + b^{2} = 0, a, b, c, d ∈ ℝ}