Vectors provide a wonderful way for us to write systems of equations compactly. You should already be familiar with two-d and three-d vectors from calculus classes. We now want to extend notions from those spaces into n-dimensional space. For example, vector addition is carried out component-wise.
The interesting new concept introduced in this section is that of span: roughly, the span of a set of vectors is the subspace generated by linear combinations of the vectors . The span represents the set of vectors that can be solutions of the system
Definition: A vector is a matrix with only a single column (``column vector''). The entries are called the components of the vector.
Note: here's a notational issue. Vectors will generally be in bold-face (on the board I'll either underline them, or overline them, depending on my mood, time of day, and what I had for breakfast). The components of named vectors are generally written with the same name, only without bold/overline/underline, and with subscripts. Notice that the components of the vector above are not at all the same as the vectors listed in the abstract, . Components are (generally) numbers....
Note: Geometrically, the sum of vectors can be found using the ``parallelogram rule'': the butt of vector is placed at the tip of the vector , and the vector from the butt of to the tip of is the sum.
Example: #4, p. 37
Q: What is the geometry of a span? What cases should be considered?
has the same solution as the linear system whose augmented matrix is
In this case, the variables - the unknowns - would be the coefficients , and a solution would consist of the appropriate possibilities of values for those coefficients.
Example: #9, p. 37
Example: #12, p. 38. In this problem we throw you for another loop, by using the letter ``a'' for vectors! You have to pay attention, and not let us mess you up too badly just by poor notation....
Example: #21, p. 38
Example: #27, p. 38