Last Name^{Name}  ^{Student No.}^{Department}  ^{Section}^{Signature} ^{:}^{:}  ^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{:}^{Code}^{Acad. Year}^{Semester}^{Date}^{Time}^{Duration}  M E T U Department of Mathematics^{Basic Linear Algebra}^{Midrem}^{Math 260}^{2001}^{Fall}^{5.1.2002}^{13:00}^{100 minutes}  ^{4 QUESTIONS ON 4 PAGES}^{TOTAL 75 POINTS}^{1}  ^{2}  ^{3}  ^{4}  ^{5}

(16 pts.) 1) Consider the vector space P_{2} of all polynomials p(x) = ax^{2} + bx + c, a, b, c ∈ IR. Determine whether each of the parts (a)-(c) defines a subspace in P_{2} (justify your answer).

a) p(0) + p(1) = 1

b) p(1) = −p(−1)

c) a = 2b = −c

d) Define an operation p ⊕ q = 2p + 3q for p, q ∈ P_{2}. Does this operation satisfy the axioms for addition in a vector space?

(24 pts.) 2) Let S = {(1, −1, 0, 0), (0, 1, −1, 0), (0, 0, 1, −1), (−1, 0, 0, 1)} ∈ IR^{4}.

a) Show that S is linearly dependent.

b) Compute the dimension of L(S) (the span of S).

c) Obtain a basis B for IR^{4} that contains the first three vectors of S.

d) Express (1, 1, 1, −3) as a linear combination of elements of the basis B in part (c).

(17 pts.) 3 a) Show that ⟨x, y⟩ = x_{1}y_{1} + x_{1}y_{2} + x_{2}y_{1} + 4x_{2}y_{2} + x_{3}y_{3} defines an inner product on IR^{3} where x = (x_{1}, x_{2}, x_{3}), y = (y_{1}, y_{2}, y_{3}).

b) Find an orthonormal basis for IR^{3} with respect to this inner product.

(18 pts.) 4. Let W be the subspace of IR^{3} spanned by the vector (2, −1, 4).

a) Find the projection of (1, 1, 0) on W.

b) Find the distance between (1, 1, 0) and W.

c) Find a basis for W^{⊥}.