Suppose \(V\) is a vector space and \(S\) is a finite spanning set for \(V\) (i.e. \(V = \Span S\)). In the previous chapter, we saw that if \(S\) is linearly dependent, then (at least) one vector can be removed from \(S\text{,}\) and the resulting smaller set will still be a spanning set. You can imagine repeating this process until finally you are left with a spanning set that is linearly independent.
This leads to the following definition.
basis for a vector space
a linearly independent spanning set for the space
Discovery18.1.
In each of the following, determine whether \(S\) is a basis for \(V\text{.}\) If it is not a basis, make sure you know which property \(S\) violates, independence or spanning.
As discussed in the introduction to this discovery guide above, a spanning set that is not linearly independent contains redundant information in the form of vectors that are not actually needed to form a spanning set. This redundancy manifests itself in other ways, as the next discovery activity will demonstrate.
Discovery18.2.
Consider the set \(S = \{(1,0),(1,1),(1,-1)\}\) of vectors in \(\R^2\text{.}\) This set spans \(\R^2\) but is not linearly independent.
(a)
Since \(S\) spans \(\R^2\text{,}\) it is possible to express vector \((3,-3)\) as a linear combination of the vectors in \(S\text{.}\)
How many different ways do you think there are to do this?
The next discovery activity will demonstrate that the redundancies of Discovery 18.2 cannot happen for a basis.
Discovery18.3.
Suppose \(V\) is a vector space, \(S = \{\uvec{v}_1,\uvec{v}_2,\uvec{v}_3\}\) is a basis for \(V\text{,}\) and \(\uvec{w}\) is a vector in \(V\text{.}\)
Since \(S\) is a spanning set, there is a way to express \(\uvec{w}\) as linear combinations of the vectors in \(S\text{:}\)
and the two different expressions for \(\uvec{w}\) above to show that having these two different expressions violates the linear independence of \(S\text{.}\)
Discovery 18.3 shows that when we have a basis \(S = \{\uvec{v}_1,\uvec{v}_2,\dotsc,\uvec{v}_n\}\) for a vector space \(V\text{,}\) each vector in \(V\) has one unique expression as a linear combination of the vectors in \(S\text{.}\) For \(\uvec{w} = c_1\uvec{v}_1 + c_2\uvec{v}_2 + \dotsb + c_n\uvec{v}_n\text{,}\) the coefficients \(c_1,c_2,\dotsc,c_n\) are called the coordinates of \(\uvec{w}\) relative to \(S\). Since these coordinates consist of \(n\) coefficients, we sometimes relate \(\uvec{w}\) to a vector in \(\R^n\) by collecting its coordinates into an \(n\)-tuple:
for the space \(\matrixring_2(\R)\) from Task 18.5.b.
(a)
In Task 18.5.b you have already determined the vector \(\uvec{w}\) in \(\matrixring_2(\R)\) that has coordinate vector \(\rmatrixOfplain{\uvec{w}}{S} = (3,-5,1,1)\text{.}\) Now do the same to determine the vector \(\uvec{v}\) in \(\matrixring_2(\R)\) that has coordinate vector \(\rmatrixOfplain{\uvec{v}}{S} = (-1,2,0,3)\text{.}\)
(b)
Do some algebra in \(\matrixring_2(\R)\text{:}\)
Using your vectors \(\uvec{v}\) from Task a, and \(\uvec{w}\) from Task 18.5.b compute the linear combination \(2 \uvec{v} + \uvec{w}\text{.}\)
Note: Vectors \(\uvec{v}\) and \(\uvec{w}\) “live” in the space \(\matrixring_2(\R)\text{,}\) so your computation in this task should involve \(2 \times 2\) matrices, and should also result in a \(2 \times 2\) matrix.
(c)
Do the same algebra in \(\R^4\text{:}\)
Compute \(2 \rmatrixOfplain{\uvec{v}}{S} + \rmatrixOfplain{\uvec{w}}{S}\text{,}\) using the coordinate vectors \(\rmatrixOfplain{\uvec{v}}{S} \) and \(\rmatrixOfplain{\uvec{w}}{S}\) provided to you in Task 18.6.a.
Note: These coordinate vectors “live” in the space \(\R^4\text{,}\) so your computation in this task should involve four-dimensional vectors, and should also result in a four-dimensional vector.
(d)
Compare your results:
Consider your four-dimensional result vector from Task c as a coordinate vector for some vector in \(\matrixring_2(\R)\) relative to \(S\text{.}\) Similarly to your computations in Task 18.5.b and Task a, determine the matrix in \(\matrixring_2(\R)\) that has coordinate vector equal to your result vector from Task c. Then compare with your result matrix from Task 18.6.b. Surprised?