# Suppose v, w ∈ Cn . An exercise similar to the one leading

Question

Assignment 6–Algebra I

Due at the beginning of tutorial Nov. 16

1. Suppose v, w ∈ Cn . An exercise similar to the one leading up to the

Cauchy-Schwarz inequality yields the equation

v+w

2

= v

2

+ 2Re( v, w ) + w 2 ,

where Re(a + bi) = a for a, b ∈ R. Take this equation for granted. It is

√

also not that diﬃcult to prove that Re(a + bi) ≤ a2 ≤ |a + bi| for every

a, b ∈ R. Using these two facts, prove the triangle inequality which states

/4

v+w ≤ v + w ,

(Hint: Use the Cauchy-Schwarz inequality to prove that the square of the

LHS is less than or equal to the square of the RHS.)

2. Let

2

−1

−4

2

v1 =

0 , v2 = 1

−4

2

−i

, v3 = 3 .

1

i

The set {v1 , v3 , v3 } is a linearly independent set in C4 (do not prove this).

Compute an orthonormal set {u1 , u2 , u3 } from it using the Gram-Schmidt

procedure. Begin the procedure with the vector v1 .

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3. Let {v1 , . . . , vp } and {u1 , . . . , up } be as in Theorem GSP in the text.

(a) Prove by strong induction on 1 ≤ k ≤ p that uk ∈ span{v1 , . . . vk }

for every 1 ≤ k ≤ p. (Hint: Have a look at the link on strong induction on the course website. The diﬀerence between strong induction

and just plain induction here is that you will make the induction

assumption of uj ∈ span{v1 , . . . , vj } for all 1 ≤ j ≤ k)

(b) Prove that uk = 0 for any 1 ≤ k ≤ p. (Hint: Look at the proof of

Theorem GSP. You will need part (a).)

SUGGESTED EXERCISES

(see next page)

• Prove the ﬁrst two facts in exercise 1.

• Compute an orthonormal set from {u1 , u2 , u3 } in exercise 2.

• All reading questions in section MO and exercises T13-T18 in section MO.

• All reading questions and exercises in section MM.

• Prove Theorem SLSLC in section LC by using Theorem SLEMM and what

you know about matrix multiplication.

1

/4

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• Choose a set of two linearly independent vectors in C3 (why is this easy?)

and apply the Gram-Schmidt procedure to the set. Normalize the resulting

orthogonal set to obtain an orthonormal set.

• Suppose {v1 , . . . , vk } is an orthonormal set in Cn and a1 , . . . , ak ∈ C.

Prove that

2

k

aj vj

k

|aj |2

=

j=1

j=1

• Give an explicit example of an orthogonal set {v1 , . . . , vk } in C2 and scalars

a1 , . . . , ak ∈ C for which the previous equation fails. (Prove both orthogonality and the failure of the equation. Hint: Take k = 2.)

• Suppose {v1 , v2 } ⊂ Cn is linearly independent. Use the Gram-Schmidt

procedure to prove that v2 = u1 + u2 where u1 is some vector orthogonal

to v1 and u2 ∈ span{v1 }.

• Prove that v2 = u1 +u2 as in the previous exercise, also when {v1 , v2 } ⊂ Cn

is linearly dependent.

• Prove that v =

n

j=1

v, ej ej . This proves that Cn = span{e1 , . . . en }.

• Use the previous exercise to prove that v ∈ Cn is orthogonal to every

vector in Cn if and only if v = 0.

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