Practice 1

Given the following matrix and it's echelon form determine a basis for and .

Problem 2

Suppose a matrix has three pivot columns. Is ? What is the dimension of ? Explain your answers.

has 3 dimensions as is rank 3 however since maps from to is a 3 dimensional subspace of and is not equal to the dimensions of via the rank theorem are meaning it lives in .

Problem 3

What is the rank of a matrix whose null space is three-dimensional?

The rank is via the rank theorem.