Linear Algebra: Orthogonality Concepts

Questions and Answers

What makes two vectors orthogonal?

If their dot product is 0.

Which of the following statements is true regarding the Fundamental Theorem of Linear Algebra Pt. 2?

Null(A) and null(A^T) are orthogonal complements.

Col(A) and null(A) are orthogonal complements. (correct)

Row(A) and col(A) are orthogonal complements.

Row(A) and null(A) are orthogonal complements. (correct)

What is the orthogonal complement of subspace V?

Contains every vector that is perpendicular to V.

What condition does an orthogonal matrix Q satisfy?

(Q^T)Q = I. Signup and view all the answers

What happens when Q is square?

(Q^T)Q = I => Q^T = Q⁻¹. Signup and view all the answers

What is the Gram-Schmidt process?

Given n independent vectors, find corresponding orthogonal vectors. Signup and view all the answers

What is the setup for the Normal Equations?

(t,b) ==> C + Dt = b. Signup and view all the answers

How is error defined in the context of regression?

e = b - p. Signup and view all the answers

What is the first step of the Gram-Schmidt process?

B = b - [(A^T)b/(A^T)A]A. Signup and view all the answers

What is the second step of the Gram-Schmidt process?

C = c - [(A^T)c/(A^T)A]A - [(B^T)c/(B^T)B]B. Signup and view all the answers

What is the A = LU (LDU) factorization?

L - lower triangular, U - upper triangular, D - only pivots in it. Signup and view all the answers

How is the dimension defined in linear algebra?

Counting independent columns. Signup and view all the answers

Study Notes

Orthogonality of Vectors

Two vectors are orthogonal if their dot product equals zero (v^Tw = 0).
The relationship ||v||² + ||w||² = ||v + w||² holds for orthogonal vectors.

Fundamental Theorem of Linear Algebra Pt. 2

The row space (row(A)) and null space (null(A)) are orthogonal complements.
The column space (col(A)) and the null space of the transpose (null(A^T)) are also orthogonal complements.

Orthogonal Complement of a Subspace

Contains all vectors that are perpendicular to the subspace V, denoted as V perp.

Orthogonal Matrix Properties

An orthogonal matrix Q satisfies the equation (Q^T)Q = I, where Q^T is the transpose of Q.

Square Orthogonal Matrix

For a square orthogonal matrix, (Q^T)Q = I implies that Q^T = Q⁻¹ (the transpose is equal to the inverse).

Gram-Schmidt Process

The Gram-Schmidt process transforms n independent vectors into a corresponding set of orthogonal vectors.
The output is represented as A = QR, where Q contains orthogonal vectors.

Normal Equations Setup

The normal equations are represented as: (t,b) ==> C + Dt = b.
Expressed in matrix form, the normal equations are (A^T)Ax° = (A^T)b.

Error in Linear Equations

The prediction p is determined by p = Ax.
The error between the actual output b and prediction p is expressed as e = b - p.

First Step of Gram-Schmidt

The first step computes B as follows: B = b - [(A^T)b/(A^T)A]A, ensuring B is orthogonal to A.

Second Step of Gram-Schmidt

The second step computes C as C = c - [(A^T)c/(A^T)A]A - [(B^T)c/(B^T)B]B, making C orthogonal to both A and B.

A = LU (LDU) Factorization

In LU factorization:
- L is a lower triangular matrix with 1's on the diagonal.
- U is an upper triangular matrix, with pivots on the diagonal.
- D consists of only the pivots.

Dimension in Linear Algebra

The dimension is calculated by counting the number of independent columns in the matrix.

Description

This quiz covers key concepts related to orthogonality in linear algebra. Topics include the definition of orthogonal vectors, properties of orthogonal matrices, the Gram-Schmidt process, and the relationships between subspaces. Test your understanding of these fundamental ideas and their implications in linear algebra.