Linear Algebra - Exam 3

NulA

What is the orthogonal complement of RowA?

NulA^T

What is the orthogonal complement of ColA?

A^-1

What is A^T equivalent to, in terms of A?

U^TU = I_n

An m x n matrix U has orthonormal columns if and only what is true

n > m

In terms of the amount of rows n and columns m, what fact means that a matrix U cannot have orthonormal columns?

(1) The columns of A are linearly independent
(2) The matrix A^TA is invertible

What are two statements that are equivalent to the statement that the equation Ax = b has a unique least-squares solution for each b within R^m

The dimension of the vector space spanned by its columns

What is the rank of a matrix

When the input vectors are linearly dependent

When will the Gram-Schmidt process fail to produce an orthogonal basis?

The # of free variables

When given the QR decomposition of A and asked to find the least-squares solution for Ax = b, what should you do?

number of free variables

What is the Dim(NulA^T) equal to?

What is the normal equation you can use to solve a least-squares problem?

1

0

It is also diagonalizable

If A is diagonalizable, then what is true of A^T?

Orthogonal matrix

What is the name of a square matrix whose columns are orthonormal?

Put it in parametric vector form

What do you do when given a matrix A and asked to find NulA?

No

Are orthonormal bases unique?

False

True or false: the matrix product of two orthogonal matrices is not necessarily an orthogonal matrix

A linear combination of v which forms the side of a triangle with u as the hypotenuse

When sketching the projection of a vector v onto u, what are you looking to draw?

Find the parametric vector form of W^T

If given the basis for a subspace W and asked to find W-perp, what should you do?

What is the quadratic equation?