Understand the equivalence between a system of linear equations and a vector equation.
Learn the definition of and how to draw pictures of spans.
Recipe: solve a vector equation using augmented matrices / decide if a vector is in a span.
Pictures: an inconsistent system of equations, a consistent system of equations, spans in and
Vocabulary word:vector equation.
Essential vocabulary word:span.
An equation involving vectors with coordinates is the same as equations involving only numbers. For example, the equation
For two vectors to be equal, all of their coordinates must be equal, so this is just the system of linear equations
A vector equation is an equation involving a linear combination of vectors with possibly unknown coefficients.
Asking whether or not a vector equation has a solution is the same as asking if a given vector is a linear combination of some other given vectors.
For example the vector equation above is asking if the vector is a linear combination of the vectors and
The thing we really care about is solving systems of linear equations, not solving vector equations. The whole point of vector equations is that they give us a different, and more geometric, way of viewing systems of linear equations.
A Picture of a Consistent System
Below we will show that the above system of equations is consistent. Equivalently, this means that the above vector equation has a solution. In other words, there is a linear combination of and that equals We can visualize the last statement geometrically. Therefore, the following figure gives a picture of a consistent system of equations. Compare with figure below, which shows a picture of an inconsistent system.
In order to actually solve the vector equation
one has to solve the system of linear equations
This means forming the augmented matrix
and row reducing. Note that the columns of the augmented matrix are the vectors from the original vector equation, so it is not actually necessary to write the system of equations: one can go directly from the vector equation to the augmented matrix by “smooshing the vectors together”. In the following example we carry out the row reduction and find the solution.
where are vectors in and are unknown scalars, has the same solution set as the linear system with augmented matrix
whose columns are the ’s and the ’s.
Now we have three equivalent ways of thinking about a linear system:
As a system of equations:
As an augmented matrix:
As a vector equation ():
The third is geometric in nature: it lends itself to drawing pictures.
It will be important to know what are all linear combinations of a set of vectors in In other words, we would like to understand the set of all vectors in such that the vector equation (in the unknowns )
has a solution (i.e. is consistent).
Let be vectors in The span of is the collection of all linear combinations of and is denoted In symbols:
We also say that is the subset spanned by or generated by the vectors
The above definition is the first of several essential definitions that we will see in this textbook. They are essential in that they form the essence of the subject of linear algebra: learning linear algebra means (in part) learning these definitions. All of the definitions are important, but it is essential that you learn and understand the definitions marked as such.
Set Builder Notation
reads as: “the set of all things of the form such that are in ” The vertical line is “such that”; everything to the left of it is “the set of all things of this form”, and everything to the right is the condition that those things must satisfy to be in the set. Specifying a set in this way is called set builder notation.
All mathematical notation is only shorthand: any sequence of symbols must translate into a usual sentence.
Three characterizations of consistency
Now we have three equivalent ways of making the same statement:
A vector is in the span of
The vector equation
has a solution.
The linear system with augmented matrix
Equivalent means that, for any given list of vectors either all three statements are true, or all three statements are false.
Pictures of Spans
Drawing a picture of is the same as drawing a picture of all linear combinations of