1. 1.3 Vector Equations

2. Geometric Interpretation?
Vectors in R2
: 1-column matrices or
ordered pairs of real numbers
Example: Given and , find
Geometric Interpretation?

3. Vectors in R3
: column matrices
or points in three-dimensional space
Vectors in Rn
: column matrices

4. Two vectors are equal if and only if their corresponding entries are equal.
The vector (where c is a real number) is a scalar multiple of .

5. Algebraic Properties of

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

7. Example: Determine whether w can be generated as a linear
Combination of v and v , where
, , and

8. 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:

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

10. Geometric Description of Span: