can you please just solve the question 1 .the question is via attachment.

 

 

Mathematics IB Tutorial 7 (week 8)
Semester 2, 2016
1. (a) (i) Let V be a subspace of Rn with orthonormal basis {v1 , . . . , vr }.
Denote by prV : Rn → Rn the linear transformation given by
orthogonal projection onto V . Prove that
prV (x) = (v1 v1t + · · · + vr vrt )x
for all x ∈ Rn .
(ii) Let A be a symmetric n × n matrix and P = (u1 · · · un ) be
a matrix that orthogonally diagonalises A, so that P t AP =
diag(λ1 , . . . , λn ). Prove that
A = λ1 u1 ut1 + · · · + λn un utn .


1 2
(b) Let A =
.
2 −2
(i) The eigenvectors of A are (1, −2) and (2, 1). Find the eigenvalue of each of these.
(ii) Check the formula in part (a) (ii) for this particular matrix.
(iii) Use this example, together with part (a) (i), to interpret the
formula in part (a) (ii) geometrically. What is it saying?
2. A pentagon with a perimeter of 90cm is to be constructed by adjoining
an equilateral triangle to a rectangle. Find the dimensions of the
rectangle and triangle that will maximise the area of the pentagon. 1

Attachments:

• 1492576604-8.pdf