Linear Algebra New Material

How to find a unit vector u in the same direction as v

For u and v in Rn, the distance between u and v

dist(u,v) == ||u-v||

When are two vectors u and v orthogonal

A vector x is in W Perp if?

1.if and only if x is orthogonal to every vector in a set that

spans W.

2. W perp is a subspace of Rn

What is the orthogonal complement of Row(A) and Col(A)

What is the connection between their inner dot product and the angle # between the two line segments from the origin to the points identified with u and v?

Suppose y is a linear combination of an orthogonal basis. How to compute c1 without row reduction

y dot u1 / u1 dot u1 = c1

an orthogonal projection is a sum of what two components

y hat: scalar * vector u
z: vector orthogonal to u

orthogonal set of unit vectors

orthonormal set

An m n matrix U has orthonormal columns if

U^Transpose * U = I

Define v1 and v2 and v3 in the Gram-Schmidt Process

How to construct an orthonormal basis given vectors v1 and v2 

Define an nxn orthogonal matrix and a least squares solution to Ax=b

(a) Least-squares solution to Ax=b

Let A be an m×n matrix and b∈Rm. A least-squares solution to Ax=b is a vector x^∈Rn that minimizes the sum of squared errors

b-Ax^ <= b-Ax for any x existing in R^n

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(b) n×n orthogonal matrix

An n×nn×n matrix QQ is orthogonal if its columns are orthonormal vectors and Q = Q^-1 and Q*Q^T = I

y^, y and z formulas