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Flashcards in Final Deck (16)
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1
Q

Theorem 1 (6.1)

A

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
2
Q

Definition of a Length of a Vector

A
4
Q

Theorem 2 (Pythagorean Theorem)

A

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

5
Q

Theorem 3 (6.1)

A

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 AT:

(Row A)perpend= Nul A and (Col A)perpd = Nul AT

6
Q

Theorem 4 (6.2)

A

If S={u1…up} 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

7
Q

Theorem 5 (6.2)

A
8
Q

Definition of Orthogonal Basis

A

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

10
Q

Theorem 8 (Orthogonal Decomposition Theorem)

A
11
Q

Theorem 9 (6.3) Best Approximate Theorem

A
12
Q

Theorem 10 (6.3)

A
13
Q

Definition of Least Squares Solution

A
14
Q

Theorem 6 (6.2)

A

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

15
Q

Theorem 14 (6.5)

A
16
Q

Theorem 15 (6.5)

A
17
Q

Theorem 7 (6.2)

A

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
22
Q

Theorem 13 (6.5)

A

The set of least-squares solutions of Ax = b coincides with the nonempty set of solutions of the normal equations ATAx = ATb