Teacher/User Name: DEMO Name:__________________________ Practice ProblemsBy Easy Worksheet Period: ________ Date: ____________

### Solve:

1) Given A is a m × n matrix. Which of the following statements are equivalent?
I. Each vector in ℜn maps onto a unique vector in ℜm
II. A has no free variables
III. The solution to Ax = 0 is always just the zero vector

By Easy Worksheet

By Easy Worksheet

By Easy Worksheet

By Easy Worksheet

2) Given A is a m × n matrix. Which of the following statements are equivalent?
I. Each column has a pivot
II. Each vector in ℜm is the image of at most one x in ℜn
III. The solution to Ax = 0 is always just the zero vector
3) Given A is a m × n matrix. Which of the following statements are equivalent?
I. T(x) is one-to-one
II. The solution to Ax = 0 is always just the zero vector
III. Each column has a pivot

By Easy Worksheet

By Easy Worksheet

By Easy Worksheet

By Easy Worksheet

4) Given A is a m × n matrix.T(x) = Ax is onto only if it has __ pivot ____