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- Math 211 (A01) Typed Homework 1

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1. Let ⃗u =

1

2

3

, ⃗v =

4

5

6

, and ⃗w =

7

8

9

. Explain whether the set A = {⃗u, ⃗v, ⃗w} is a basis for R

3

.

Make sure to include all relevant definitions.

2. Fix ⃗u, ⃗v ∈ R

n. Show that span(span{⃗u, ⃗v}) = span{⃗u, ⃗v}. Make sure to include all relevant

definitions.

3. The worksheets define proj⃗v⃗u as the vector in the direction ⃗v such that ⃗u−proj⃗v⃗u is orthogonal

to ⃗v. Call this definition (a). Your textbook defined proj⃗v⃗u as the vector ⃗u·⃗v

⃗v·⃗v ⃗v. Call this

definition (b). Show that definitions (a) and (b) are equivalent by showing that that the

vector arising from definition (b) must be the same as the vector arising from definition (a).

In your answer, elaborate on definition (a) by including the definition of vector in the direction

of ⃗v and orthogonal.

1

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