TBIL-LA Linear Independence (EV4)

We say that a set of vectors is linearly dependent if one vector in the set belongs to the span of the others. Otherwise, we say the set is linearly independent.

Figure 14. A linearly dependent set of three vectors

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You can think of linearly dependent sets as containing a redundant vector, in the sense that you can drop a vector out without reducing the span of the set. In the above image, all three vectors lay in the same planar subspace, but only two vectors are needed to span the plane, so the set is linearly dependent.

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Activity 2.4.4.

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Consider the following three vectors in

R^{3} :

{\vec{v}}_{1} = [\begin{matrix} - 2 \\ 0 \\ 0 \end{matrix}], {\vec{v}}_{2} = [\begin{matrix} 1 \\ 3 \\ 0 \end{matrix}], and {\vec{v}}_{3} = [\begin{matrix} - 2 \\ 5 \\ 4 \end{matrix}] .

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(a)

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Let

\vec{w} = 3 {\vec{v}}_{1} - {\vec{v}}_{2} - 5 {\vec{v}}_{3} = [\begin{matrix} ? \\ ? \\ ? \end{matrix}] .

The set

{{\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}, \vec{w}}

is...

linearly dependent: at least one vector is a linear combination of others
linearly independent: no vector is a linear combination of others

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(b)

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Find

RREF [\begin{array}{cccc} {\vec{v}}_{1} & {\vec{v}}_{2} & {\vec{v}}_{3} & \vec{w} \end{array}] = RREF [\begin{array}{cccc} - 2 & 1 & - 2 & ? \\ 0 & 3 & 5 & ? \\ 0 & 0 & 4 & ? \end{array}] = ? .

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What does this tell you about solution set for the vector equation

x_{1} {\vec{v}}_{1} + x_{2} {\vec{v}}_{2} + x_{3} {\vec{v}}_{3} + x_{4} \vec{w} = \vec{0} ?

It is inconsistent.
It is consistent with one solution.
It is consistent with infinitely many solutions.

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(c)

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Which of these might explain the connection?

A pivot column establishes linear independence and creates a contradiction.
A non-pivot column both describes a linear combination and reveals the number of solutions.
A pivot row describes the bound variables and prevents a contradiction.
A non-pivot row prevents contradictions and makes the vector equation solvable.

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Fact 2.4.5.

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For any vector space, the set

{{\vec{v}}_{1}, \dots {\vec{v}}_{n}}

is linearly dependent if and only if the vector equation

x_{1} {\vec{v}}_{1} + x_{2} {\vec{v}}_{2} + \dots + x_{n} {\vec{v}}_{n} = \vec{0}

is consistent with infinitely many solutions.

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Likewise, the set of vectors

{{\vec{v}}_{1}, \dots {\vec{v}}_{n}}

is linearly independent if and only the vector equation

x_{1} {\vec{v}}_{1} + x_{2} {\vec{v}}_{2} + \dots + x_{n} {\vec{v}}_{n} = \vec{0}

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has exactly one solution:

[\begin{matrix} x_{1} \\ ⋮ \\ x_{n} \end{matrix}] = [\begin{matrix} 0 \\ ⋮ \\ 0 \end{matrix}] .

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Activity 2.4.6.

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Find

RREF [\begin{array}{cccccc} 2 & 2 & 3 & - 1 & 4 & 0 \\ 3 & 0 & 13 & 10 & 3 & 0 \\ 0 & 0 & 7 & 7 & 0 & 0 \\ - 1 & 3 & 16 & 14 & 1 & 0 \end{array}]

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and mark the part of the matrix that demonstrates that

S = {[\begin{matrix} 2 \\ 3 \\ 0 \\ - 1 \end{matrix}], [\begin{matrix} 2 \\ 0 \\ 0 \\ 3 \end{matrix}], [\begin{matrix} 3 \\ 13 \\ 7 \\ 16 \end{matrix}], [\begin{matrix} - 1 \\ 10 \\ 7 \\ 14 \end{matrix}], [\begin{matrix} 4 \\ 3 \\ 0 \\ 1 \end{matrix}]}

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is linearly dependent (the part that shows its linear system has infinitely many solutions).

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Activity 2.4.7.

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(a)

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Write a statement involving the solutions of a vector equation that’s equivalent to each claim:

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(i)

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“The set of vectors

{[\begin{matrix} 1 \\ - 1 \\ 0 \\ - 1 \end{matrix}], [\begin{matrix} 5 \\ 5 \\ 3 \\ 1 \end{matrix}], [\begin{matrix} 9 \\ 11 \\ 6 \\ 2 \end{matrix}]}

is linearly independent.”

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(ii)

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“The set of vectors

{[\begin{matrix} 1 \\ - 1 \\ 0 \\ - 1 \end{matrix}], [\begin{matrix} 5 \\ 5 \\ 3 \\ 1 \end{matrix}], [\begin{matrix} 9 \\ 11 \\ 6 \\ 2 \end{matrix}]}

is linearly dependent.”

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(b)

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Explain how to determine which of these statements is true.

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Observation 2.4.8.

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Compare the following results:

A set of $R^{m}$ vectors ${{\vec{v}}_{1}, \dots {\vec{v}}_{n}}$ is linearly independent if and only if $RREF [\begin{array}{ccc} {\vec{v}}_{1} & \dots & {\vec{v}}_{n} \end{array}]$ has all pivot columns.
A set of $R^{m}$ vectors ${{\vec{v}}_{1}, \dots {\vec{v}}_{n}}$ is linearly dependent if and only if $RREF [\begin{array}{ccc} {\vec{v}}_{1} & \dots & {\vec{v}}_{n} \end{array}]$ has at least one non-pivot column.
A set of $R^{m}$ vectors ${{\vec{v}}_{1}, \dots {\vec{v}}_{n}}$ spans $R^{m}$ if and only if $RREF [\begin{array}{ccc} {\vec{v}}_{1} & \dots & {\vec{v}}_{n} \end{array}]$ has all pivot rows.
A set of $R^{m}$ vectors ${{\vec{v}}_{1}, \dots {\vec{v}}_{n}}$ fails to span $R^{m}$ if and only if $RREF [\begin{array}{ccc} {\vec{v}}_{1} & \dots & {\vec{v}}_{n} \end{array}]$ has at least one non-pivot row.

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Activity 2.4.9.

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What is the largest number of

R^{4}

vectors that can form a linearly independent set?

$3$
$4$
$5$
You can have infinitely many vectors and still be linearly independent.

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Activity 2.4.10.

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Is it possible for the set of Euclidean vectors

{{\vec{v}}_{1}, {\vec{v}}_{2}, \dots, {\vec{v}}_{n}, \vec{0}}

to be linearly independent?

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Subsection 2.4.3 Individual Practice

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Remark 2.4.11.

Recall that in Activity 2.2.1 we used the words vector, linear combination, and span to make an anology with recipes, ingredients, and meals. In this analogy, a recipe was defined to be a list of amounts of each ingredient to build a particular meal.

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