Name:
Winter 2006 Quiz # 1 ECE S-510
You are given the following two vectors r1 = [1 2]t and r2 = [2 1]t.
(a) Can you span the R2 space using r1 and r2? Why?
(b) Find ||r1|| and ||r2||. Are r1 and r2 orthogonal? Why?
(c) If you are given the vector r = [3 1]t , can you express r as a linear combination
of r1 and r2, i.e., r = C1 r1 + C2 r2? Why? Find C1 and C2. Why aren’t C1 and C2
just the respective projections of r onto r1 and r2?
(d) If we want to span the R2 space using two orthonormal vectors
φ
1 and
φ
2 such
that r1 is in the direction of
φ
1. Find
φ
1 and
φ
2 in terms of r1 and r2.
(e) If we want to approximate r in part (c) by either r1 or r2, which one would you
use and why?
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