Basis

Section 4.2 Basis

Goals.

We will

extend the concept of linear independence to general vector spaces,
extend the concept of span to general vectors spaces, and
combine independence with span to construct vector spaces from a minimal, sufficient subset of vectors

Subsection 4.2.1 Independence

Our goal here is to extend the concept of dependent/independent from vectors in \(\R^n\) (vectors from previous classes) to vectors from any space. Review Definition 1.4.2.

Checkpoint 4.2.1.

Determine if the following sets are independent by inserting them into the definition of independent. As necessary ignore that these vectors are not in \(\R^n\text{.}\) You will still know how to do the arithmetic. Do not look for shortcuts: that will take longer than the simple arithmetic.

(a)

\([2,3,9],\) \([1,0,9],\) \([1,3,7]\)

(b)

\(\left[ \begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array} \right],\) \(\left[ \begin{array}{rr} 2 & 3 \\ 1 & 0 \end{array} \right],\) \(\left[ \begin{array}{rr} 8 & 13 \\ 3 & 2 \end{array} \right]\)

(c)

\(x^2+6x+3,\) \(x^2+1,\) \(3x+1\)

Next we look at some related examples and use the results to notice a truth about dependent sets. Make sure you see the connection between these sets of vectors and those of the previous checkpoint.

Checkpoint 4.2.2.

Determine if the following sets are independent by inserting them into the definition of independent.

(a)

\([2,3,9],\) \([1,0,9]\)

(b)

\(\left[ \begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array} \right],\) \(\left[ \begin{array}{rr} 8 & 13 \\ 3 & 2 \end{array} \right]\)

(c)

\(x^2+6x+3,\) \(3x+1\)

Checkpoint 4.2.3.

(a)

What was the difference between the sets of vectors in Checkpoint 4.2.1 and Checkpoint 4.2.2 ?

(b)

Contrasting the results, what is suggested about subsets with regards to dependence/independence?

(c)

If a set is dependent, how can we change the set to produce an independent set?

Subsection 4.2.2 Span and Dependence

Review Definition 1.2.3. We will extend it here to the span of vectors in any vector space and use it to determine some general statements about dependent/independent sets.

First, we want to note the connection between span and depdendent. If there is a non-trivial solution to \(a\vec{v}_1+b\vec{v}_2+c\vec{v}_3=\vec{0}\) then it can be re-arranged into \(\vec{v}_1=\frac{b}{a}\vec{v}_2+\frac{c}{a}\vec{v}_3\text{.}\) Thus if set of vectors is dependent, the first vector is in the span of the other two. Actually, any vector in the dependent set is in the span of the others (why?).

Checkpoint 4.2.4.

(a)

\(2x^2+2x\) is dependent on \(\{x^2,x,1\}\text{.}\) Write it as a linear combination of some of these.

(b)

\(2x^2+2x+3=2(x^2+x)+3(1)\text{.}\) Re-arrange this to show \(2x^2+2x+3\) is in the span of \(\{x^2,x,1\}\text{.}\)

(c)

With this in mind will the span of \(\{x^2+x,x^2,x,1\}\) contain vectors not in the span of \(\{x^2,x,1\}\text{?}\)

(d)

If dependent vectors are removed from a set, will it change the span?

(e)

How can one find the minimal, spanning set of a vector space?

Checkpoint 4.2.5.

Explain each of the following.

(a)

If \(\{\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_n\}\) is an independent set, then any subset is an independent set.

(b)

If \(\{\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_n\}\) is a dependent set, then any superset is a dependent set.

(c)

Prove or disprove that any subset of a dependent set is dependent.

Subsection 4.2.3 Basis

Combining the concepts of span and independent using the conclusions of this section leads us to the following concept.

Definition 4.2.6.

A set of vectors \(\{\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_n\}\) is a basis for a vector space \(H\) if and only if the set is linearly independent and \(\text{span}\{\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_n\}=H\text{.}\)

We can think of a basis as the smallest set of vectors that can be used to produce the whole set using only linear combinations.

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