Math 260 Basic Linear Algebra Midterm Exam II (Spring 1999-2000)

1 / (25)  2 / (25)  3 / (25)  4 / (25)  TOTAL

4 questions on 4 pages Total 100 points

Name and Surname: Student No: Section: Signature: Duration: 100 minutes

1. Let P_{1} be the vector space of polynomials of degree ≤ 1 over ℝ in the indeterminate t.

If it is possible, determine the scalars a and b such that (p | q) = a ∫_{0}^{1} p(t)q(t)dt − bq(0)

defines an inner product on P_{1} and ||t|| = 1.

2. Let A = [ \begin{matrix}1 & 1 & 2 & 1 & 0 \\ 0 & 0 & 0 & 3 & −3 \\ 1 & 1 & 2 & 2 & −1 \end{matrix} ]

a) Find a basis for the column space of A.

b) Find a basis for the row space of A.

c) Find the dimension of S = {X ∈ ℝ^{⋬×⊮} | 𝔸𝕏 = ⊬} 3. Let {u_{1}, u_{2}, u_{3}, u_{4}} be an orthonormal basis for a real inner product space V, and let W be

a subspace of V whose basis is B = {v_{1} = u_{1}, v_{2} = u_{1}+u_{2}+u_{3}, v_{3} = u_{1}+u_{2}−u_{3}+u_{4}}.

a) Find an orthogonal basis for W.

b) Find an orthogonal basis for W^{⊥}.

c) Let v ∈ W^{⊥}. What is the orthogonal projection of v on W.

4. a) Show that the functions f(x) = x, g(x) = x^{2}, h(x) = e^{x} defined on ℝ are linearly independent.

b) Let p(x) = 5x^{4} − 7x^{3} − 10x + 27 and B = {1, x, x^{2}, x^{3}, x^{4}} be an ordered basis for P_{4}. Write the coordinate matrix of p(x) with respect to B.

c) Let C = {1, 1 + x, 3 + x^{2}} be a given ordered basis for a subspace of P_{2}. Suppose q(x)

has coordinate matrix [ \begin{matrix}3 \\ 5 \\ 8 \end{matrix} ] with respect to C. Write the polynomial q(x) explicitly.