Prepared materials for Day 4, Feb 9, 2018

# 1.4 Ax = b

## Theorem 4

A is an m-by-n matrix.

b is a vector in *R*^{m}.

The following 4 statements are equivalent:

- Ax = b has a solution for every b.
- Every b in
*R*^{m} is a linear combination of the columns of A.
- The columns of A span
*R*^{m}.
- A has a pivot position in every row.

## 1.4 Example 3

If not, describe the set of b values (if any) for which a solution does exist.

## 1.4.37

Do the columns of this matrix A span *R*^{4}?

# 1.5 Homogeneous systems

Ax = 0

## 1.5.7

Describe the solution set of Ax = 0 for this A:

For inhomogeneous problem will need the RRE form.

# 1.10.5

Solve for the loop currents