Final

Question 1

Theorem 1 (6.1)

Answer 1

Let u, v, and w be vectors in Rⁿ, and let c be a scalar. Then

1. u * v = v * u (dot product)
2. (u+v)*w = u*w + v*w
3. (cu)*v = c*(uv) = u*(cv)
4. u*u >/= 0, and u*u=0 if and only if u = 0

Question 2

Definition of a Length of a Vector

Answer 2

Question 4

Theorem 2 (Pythagorean Theorem)

Answer 4

Two vectors u and v are orthogonal if and only if
||u + v||² = ||u||² + ||v||²

Question 5

Theorem 3 (6.1)

Answer 5

Let A be an m x n matrix. The orthogonal complement of the row space of A is the null space of A, and the orthogonal complement of the column space of A is the null space of A^T:

(Row A)^perpend= Nul A and (Col A)^perpd = Nul A^T

Question 6

Theorem 4 (6.2)

Answer 6

If S={u_1…u_p} is an orthogonal set of nonzero vectors in Rⁿ then S is linearly independent and hence is a basis for the subspace spanned by S

Question 7

Theorem 5 (6.2)

Answer 7

Question 8

Definition of Orthogonal Basis

Answer 8

An orthogonal basis for a subspace W of Rⁿ is a basis for W that is also an orthogonal set.

Question 10

Theorem 8 (Orthogonal Decomposition Theorem)

Answer 10

Question 11

Theorem 9 (6.3) Best Approximate Theorem

Answer 11

Question 12

Theorem 10 (6.3)

Answer 12

Question 13

Definition of Least Squares Solution

Answer 13

Question 14

Theorem 6 (6.2)

Answer 14

An m x n matrix U has orthonormal columns if and only if U^TU = I

Question 15

Theorem 14 (6.5)

Answer 15

Question 16

Theorem 15 (6.5)

Answer 16

Question 17

Theorem 7 (6.2)

Answer 17

Let U be an m x n matrix with orthonormal columns and let x and y be in Rⁿ

1. ||Ux|| = ||x||
2. (Ux)*(Uy) = x*y (dot product)
3. (Ux)*(Uy) = 0 if and only if x*y=0

Question 22

Theorem 13 (6.5)

Answer 22

The set of least-squares solutions of Ax = b coincides with the nonempty set of solutions of the normal equations A^TAx = A^Tb