Linear Algebra Midterm 2

What is the rank of an m x n matrix A?

the dimension of the column space

What is the column space of an m x n matrix A?

the set of all linear combinations of the columns of A

What is the nullity of A?

The dimension of the null space

What is the null space of an m x n matrix A?

The set of all solutions of the homogeneous equation Ax = 0

What does Ax = 0 mean?

A: coefficient matrix

x: is the vector of variables

0: the zero vector

What is the Rank Theorem?

rank A + nullity A = number of columns in A

OR

dim(r(t)) + dim(ker(t)) = dim(dom)

What is one-to-one?

A mapping T: Rⁿ => R^m is said to be one-to-one if each b in R^m is the image of AT MOST ONE x in Rⁿ

What is onto?

A mapping T: Rⁿ => R^m is said to be onto R^m if each b in R^m is the image of AT LEAST ONE x in Rⁿ