Identifying a Basis (V6)

Section 2.6 Identifying a Basis (V6)

Definition 2.6.1.

A basis is a linearly independent set that spans a vector space.

The standard basis of $R^{n}$ is the set ${{\vec{e}}_{1}, \dots, {\vec{e}}_{n}}$ where

\begin{aligned} {\vec{e}}_{1} & = [\begin{array}{c} 1 \\ 0 \\ 0 \\ ⋮ \\ 0 \\ 0 \end{array}] & {\vec{e}}_{2} & = [\begin{array}{c} 0 \\ 1 \\ 0 \\ ⋮ \\ 0 \\ 0 \end{array}] & \dots & {\vec{e}}_{n} = [\begin{array}{c} 0 \\ 0 \\ 0 \\ ⋮ \\ 0 \\ 1 \end{array}] . \end{aligned}

For $R^{3},$ these are the vectors ${\vec{e}}_{1} = \hat{ı} = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}], {\vec{e}}_{2} = \hat{ȷ} = [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}],$ and ${\vec{e}}_{3} = \hat{k} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}] .$

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Observation 2.6.2.

A basis may be thought of as a collection of building blocks for a vector space, since every vector in the space can be expressed as a unique linear combination of basis vectors.

For example, in many calculus courses, vectors in $R^{3}$ are often expressed in their component form

(3, - 2, 4) = [\begin{matrix} 3 \\ - 2 \\ 4 \end{matrix}]

or in their standard basic vector form

3 {\vec{e}}_{1} - 2 {\vec{e}}_{2} + 4 {\vec{e}}_{3} = 3 \hat{ı} - 2 \hat{ȷ} + 4 \hat{k} .

Since every vector in $R^{3}$ can be uniquely described as a linear combination of the vectors in ${{\vec{e}}_{1}, {\vec{e}}_{2}, {\vec{e}}_{3}},$ this set is indeed a basis.

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Activity 2.6.3.

Label each of the sets $A, B, C, D, E$ as

SPANS $R^{4}$ or DOES NOT SPAN $R^{4}$
LINEARLY INDEPENDENT or LINEARLY DEPENDENT
BASIS FOR $R^{4}$ or NOT A BASIS FOR $R^{4}$

by finding $RREF$ for their corresponding matrices.

\begin{aligned} A & = {[\begin{array}{c} 1 \\ 0 \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 1 \\ 0 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 1 \\ 0 \end{array}], [\begin{array}{c} 0 \\ 0 \\ 0 \\ 1 \end{array}]} & B & = {[\begin{array}{c} 2 \\ 3 \\ 0 \\ - 1 \end{array}], [\begin{array}{c} 2 \\ 0 \\ 0 \\ 3 \end{array}], [\begin{array}{c} 4 \\ 3 \\ 0 \\ 2 \end{array}], [\begin{array}{c} - 3 \\ 0 \\ 1 \\ 3 \end{array}]} \\ C & = {[\begin{array}{c} 2 \\ 3 \\ 0 \\ - 1 \end{array}], [\begin{array}{c} 2 \\ 0 \\ 0 \\ 3 \end{array}], [\begin{array}{c} 3 \\ 13 \\ 7 \\ 16 \end{array}], [\begin{array}{c} - 1 \\ 10 \\ 7 \\ 14 \end{array}], [\begin{array}{c} 4 \\ 3 \\ 0 \\ 2 \end{array}]} & D & = {[\begin{array}{c} 2 \\ 3 \\ 0 \\ - 1 \end{array}], [\begin{array}{c} 4 \\ 3 \\ 0 \\ 2 \end{array}], [\begin{array}{c} - 3 \\ 0 \\ 1 \\ 3 \end{array}], [\begin{array}{c} 3 \\ 6 \\ 1 \\ 5 \end{array}]} \\ E & = {[\begin{array}{c} 5 \\ 3 \\ 0 \\ - 1 \end{array}], [\begin{array}{c} - 2 \\ 1 \\ 0 \\ 3 \end{array}], [\begin{array}{c} 4 \\ 5 \\ 1 \\ 3 \end{array}]} \end{aligned}

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Activity 2.6.4.

If ${{\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}, {\vec{v}}_{4}}$ is a basis for $R^{4},$ that means $RREF [{\vec{v}}_{1} {\vec{v}}_{2} {\vec{v}}_{3} {\vec{v}}_{4}]$ doesn't have a non-pivot column, and doesn't have a row of zeros. What is $RREF [{\vec{v}}_{1} {\vec{v}}_{2} {\vec{v}}_{3} {\vec{v}}_{4}] ?$

RREF [{\vec{v}}_{1} {\vec{v}}_{2} {\vec{v}}_{3} {\vec{v}}_{4}] = [\begin{array}{cccc} ? & ? & ? & ? \\ ? & ? & ? & ? \\ ? & ? & ? & ? \\ ? & ? & ? & ? \end{array}]

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Fact 2.6.5.

The set ${{\vec{v}}_{1}, \dots, {\vec{v}}_{m}}$ is a basis for $R^{n}$ if and only if $m = n$ and $RREF [{\vec{v}}_{1} \dots {\vec{v}}_{n}] = [\begin{array}{cccc} 1 & 0 & \dots & 0 \\ 0 & 1 & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & 1 \end{array}] .$

That is, a basis for $R^{n}$ must have exactly $n$ vectors and its square matrix must row-reduce to the so-called identity matrix containing all zeros except for a downward diagonal of ones. (We will learn where the identity matrix gets its name in a later module.)

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Subsection 2.6.1 Videos

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Figure 2.6.6. Video: Verifying that a set of vectors is a basis of a vector space

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Exercises 2.6.2 Exercises

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Exercises available at checkit.clontz.org ¹.

https://checkit.clontz.org/#/banks/tbil-la/V6/