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 Rm.

The following 4 statements are equivalent:

  • Ax = b has a solution for every b.
  • Every b in Rm is a linear combination of the columns of A.
  • The columns of A span Rm.
  • A has a pivot position in every row.

1.4 Example 3

1.4_example_3.png

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 R4?

1.4.37.png

1.5 Homogeneous systems

Ax = 0

1.5.7

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

1.5.7b.png

For inhomogeneous problem will need the RRE form.

1.10.5

Solve for the loop currents

1.10.5.png