Ch 5.5 Quizlet

1c. Does sets of vectors form an orthonormal basis for R2? {(1, −1)T , (1, 1)T }

No

Parseval's Formula

||v||^2 = <v,v>

orthonormal basis

A basis that is an orthogonal set of unit vectors. That is, the number of vectors that form the orthogonal set V, n = dim V

Orthonormal set of vectors

A set of vectors V that are both 1- Orthogonal (inner product is 0) 2- Unit vectors (magnitude is 1)

Orthogonal Sets of Vectors

A set of vectors, V, such that if you take the inner product of any 2 vectors in the set, their inner product <v1, v2>=0

1b. Does sets of vectors form an orthonormal basis for R2? { (3/5 , 4/5)^T , (5/13 , 12/13)^T }

No

2b. u1 = (1/(3√2) 1/(3√2) −4/3√2) u2 = (2/3 2/3 1/3) u3 = (1/√2 −1/√2 0) Let x = (1, 1, 1)T . Write x as a linear combination of u1, u2, and u3 using Theorem 5.5.2 and use Parseval's formula to compute ||x||.

Since x=−2/3√2u1+5/3u2+0u3we have by Parseval's Formula that ∥x∥^2=(−2/3√2)^2+(5/3)^2+0^2=2/9+25/9+0 and then ∥x∥=√3

12) If Q is an n × n orthogonal matrix and x and y are nonzero vectors in Rn, then how does the angle between Qx and Qy compare with the angle between x and y? Prove your answer.

The angles are equal

What is the inverse of a orthogonal matrix?

The matrix transposed (columns become rows, rows become columns)

7. Let {u1, u2, u3} be an orthonormal basis for an inner product space V. If x = c1u1 + c2u2 + c3u3 is a vector with the properties ||x|| = 5, <u1, x> = 4, and x is orthogonal to u2, then what are the possible values of c1, c2, c3?

Theorem 5.5.2: Let {u1, u2, ... , un} be an orthonormal basis for an inner product space V. If v = n∑i=1 ciui, then ci = <v, ui>.

1a. Does sets of vectors form an orthonormal basis for R2? {(1, 0)T , (0, 1)T }

Yes

1d. Does sets of vectors form an orthonormal basis for R2? {(√3/2 , 1/2 )T , (−1/2 , √3/2 )^T }

Yes

If {v1, v2, ... , vn} is an orthogonal set of nonzero vectors in an inner product space V, then v1, v2, ... , vn are...

linearly independent

How to find <u,v>

take the sum of each entry: u1v1 + u2v2 + u3v3... unvn

orthogonal matrix

the column vectors of the matrix form an orthogonal set

Let {u1, u2, ... , un} be an orthonormal basis for an inner product space V. Then and v e V can be represented by...

v=c1u1 + c2u2 +...+cnvn where V is a linear combination of u1, u2... un and c1, c2...cn are called "observations"