Can you please help me on #2,3 for "final-written", and #11-20 for "final-quiz" just the answer please please?

Math 636 Final - Written Component 1. Find a and b to obtain the best fitting equation of the form y = a + bt2 for the given
data.
t −2 −1 0 1
2
y −2 0 1 −1 −3
2. Assume that A is an m × n matrix and B is an n × m matrix where m 6= n. Prove that
if AB = Im , then rank(B) = m.
3. Prove Corollary 6.3.4 which states: If A is an n × n matrix with distinct eigenvalues
λ1 , . . . , λk , then A is diagonalizable if and only if gλi = aλi for 1 ≤ i ≤ k.
4. Let W be a subspace of a finite dimensional inner product space V. Prove that projW (~v )
is independent of basis. That is, prove that if {~v1 , . . . , ~vk } and {w
~ 1, . . . , w
~ k } are
both orthogonal bases for W, then
h~v , ~vk i
h~v , w
~ 1i
h~v , w
~ ki
h~v , ~v1 i
~v1 + · · · +
~vk =
w
~1 + · · · +
w
~k
2
2
2
k~v1 k
k~vk k
kw
~ 1k
kw
~ k k2
NOTE: Notice that our definition of projW is dependent on the choice of orthogonal basis.
So, you cannot assume that projW (~v ) = projW (~v ) if we are using different bases. This is
what you are supposed to prove in this question. 1

Attachments:

• 1492751566-final-written.pdf
• 1492751566-final-quiz.pdf