$30.00
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.
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