1. REVIEW Linear Combinations Given vectors and given scalars
is a linear combination of with weights
Example:

2. REVIEW A vector equation
has the same solution set as the linear system whose
augmented matrix is
can be generated by a linear combination of vectors in
if and only if the following linear system is consistent:

3. REVIEW
Definition
If , then the set of all linear combinations of is denoted by
Span and is called the
subset of spanned by

4. REVIEW
Example:

5. 1.4 The Matrix Equation

6. Key Idea
We will see how think of a linear combination of vectors as a product of a matrix and a vector.

7. Linear Combinations
is a linear combination of the columns of A with corresponding entries in x as weights.
Note: is defined only if the number of columns of A equals the number of elements in .

8. Example:

9. Compute

10. A matrix equation
has the same solution set as the vector equation
which has the same solution set as the linear system
whose augmented matrix is
Therefore:
Ax = b has a solution if and only if
b is a linear combination of columns of A

11. The following statements are equivalent:
Theorem 4:
The following statements are equivalent:
For each vector b, the equation has a solution.
2. Each vector b is a linear combination of the columns of A.
3. The columns of A span
4. A has a pivot position in every row.
Note: Theorem 4 is about a coefficient matrix A, not an augmented matrix.

12. Row-Vector Rule for Computing
If the product is defined, then the ith entry in is the sum of the products of corresponding entries from row i of A and from the vector x.

13. Theorem 5: