Identify the independent columns of a use the definition of

Question

1. Given

	1	2	5	4
A =	1	3	6	3
	2	4	10	8

Find each of the following:

Solvability condition

Particular solution for

	1
b =	2
	2

Complete solution

2. Choose a vector in each of the fundamental spaces of A. Prove that your choice lives in the space the use these to verify the following.

∀b ∈ C(A) ⊥ ∀y ∈ N(A^T)

and

∀r ∈ C(A^T) ⊥ ∀x ∈ N(A)

3. Identify the independent columns of A. Use the definition of linear independence to prove that the independent columns of A are linearly independent.

4. In the vector space P₃ of all p (x) = a₀ + a₁x + a₂x² + a₃x³, let S = {p(x)|p(x)∈P₃,p'(0)= 0]. Verify that S is a subspace.

5. Determine if T(v) v except that T(0, v₂) = (0,0) is a linear transformation.

6. Prove that two nonzero orthogonal vectors in R² are linearly independent.

7. What multiple of

	1
A =	1
	1

is closest to the point b =

	1
b =	2
	2

8. Find the projection matrix P that projects b onto the line through a.