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Problem 4 Let v1 = 1 2i and v2 = 4 −2i , and let ⟨x, y⟩ be the standard complex inner product on C2 . 1. Show that v1 ⊥ v2. 2. Normalize v1 and v2 to produce orthonormal vectors q1 = v1 \| v1\| ; q2 = v2 \| v2\| . 3. You are given the vector b = 7 3 . Find the coefficients c1 and c2 such that b = c1q1 + c2q2. 4. You are given the vector b = i −1 . Find the coefficients c1 and c2 such that b = c1q1 + c2q2. Hint: In parts 3 and 4, use the orthonormality of q1 and q2.

          Problem 4 Let v1 =
1
2i
and v2 =
4
−2i
, and let ⟨x, y⟩ be the standard complex inner product
on C2
.
1. Show that v1 ⊥ v2.
2. Normalize v1 and v2 to produce orthonormal vectors
q1 =
v1
\| v1\| 
; q2 =
v2
\| v2\| 
.
3. You are given the vector b =
7
3
. Find the coefficients c1 and c2 such that b = c1q1 + c2q2.
4. You are given the vector b =
i
−1
. Find the coefficients c1 and c2 such that b = c1q1 +
c2q2.
Hint: In parts 3 and 4, use the orthonormality of q1 and q2.

Added by Danny A.

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