TBIL-LA Linear Independence (VS5)

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Section 2.5 Linear Independence (VS5)

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Learning Outcomes

Determine if a set of Euclidean vectors is linearly dependent or independent by solving an appropriate vector equation.

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Subsection 2.5.1 Class Activities

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Activity 2.5.1.

Consider the two sets

S = {[\begin{matrix} 2 \\ 3 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 1 \\ 4 \end{matrix}]} T = {[\begin{matrix} 2 \\ 3 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ 1 \\ 4 \end{matrix}], [\begin{matrix} - 1 \\ 0 \\ - 11 \end{matrix}]} .

Which of the following is true?

$span S$ is bigger than $span T .$
$span S$ and $span T$ are the same size.
$span S$ is smaller than $span T .$

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Definition 2.5.2.

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 16. A linearly dependent set of three vectors

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 2.5.3.

Begin with 3 vectors in $R^{3}$

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

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(a)

Choose three non-zero scalars, $a, b,$ and $c .$ Let $\vec{w} = a {\vec{v}}_{1} + b {\vec{v}}_{2} + c {\vec{v}}_{3} .$ Is the set ${{\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}, \vec{w}}$ linearly dependent?

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(b)

Find

RREF [\begin{array}{cccc} ⋮ & ⋮ & ⋮ & ⋮ \\ {\vec{v}}_{1} & {\vec{v}}_{2} & {\vec{v}}_{3} & \vec{w} \\ ⋮ & ⋮ & ⋮ & ⋮ \end{array}] .

What does this tell you about solution set for the vector equation $x_{1} {\vec{v}}_{1} + x_{2} {\vec{v}}_{2} + x_{3} {\vec{v}}_{3} + x_{4} \vec{w} = \vec{0} ?$

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Activity 2.5.4.

Let ${\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}$ be vectors in $R^{n} .$ Suppose $3 {\vec{v}}_{1} - 5 {\vec{v}}_{2} = {\vec{v}}_{3},$ so the set ${{\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}}$ is linearly dependent. Which of the following is true of the vector equation $x_{1} {\vec{v}}_{1} + x_{2} {\vec{v}}_{2} + x_{3} {\vec{v}}_{3} = \vec{0} ?$

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

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Fact 2.5.5.

For any vector space, the set ${{\vec{v}}_{1}, \dots {\vec{v}}_{n}}$ is linearly dependent if and only if the vector equation $x_{1} {\vec{v}}_{1} + \dots + x_{n} {\vec{v}}_{n} = \vec{0}$ is consistent with infinitely many solutions.

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Activity 2.5.6.

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}]

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}]}

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

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Observation 2.5.7.

A set of Euclidean 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 a column without a pivot position.

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Observation 2.5.8.

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 ${{\vec{v}}_{1}, \dots {\vec{v}}_{n}}$ 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 ${{\vec{v}}_{1}, \dots {\vec{v}}_{n}}$ is linearly independent if and only the vector equation

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

has exactly one solution, with $c_{1} = c_{2} = \dots = c_{n} = 0 .$

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Activity 2.5.9.

Consider whether the set of Euclidean vectors ${[\begin{matrix} - 4 \\ 2 \\ 3 \\ 0 \\ - 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 0 \\ 0 \\ 3 \end{matrix}], [\begin{matrix} 1 \\ 10 \\ 10 \\ 2 \\ 6 \end{matrix}], [\begin{matrix} 3 \\ 4 \\ 7 \\ 2 \\ 1 \end{matrix}]}$ is linearly dependent or linearly independent.

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(a)

Reinterpret this question as an appropriate question about solutions to a vector equation.

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(b)

Use the solution to this question to answer the original question.

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Activity 2.5.10.

Consider whether the set of polynomials ${x^{3} + 1, x^{2} + 2 x, x^{2} + 7 x + 4}$ is linearly dependent or linearly independent.

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(a)

Reinterpret this question as an appropriate question about solutions to a polynomial equation.

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(b)

Use the solution to this question to answer the original question.

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Activity 2.5.11.

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 2.5.12.

What is the largest number of

P_{4} = {a x^{4} + b x^{3} + c x^{2} + d x + e | a, b, c, d, e \in R}

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 2.5.13.

What is the largest number of

P = {f (x) | f (x) is any polynomial}

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 2.5.14.

Is is possible for the set of vectors ${{\vec{v}}_{1}, {\vec{v}}_{2}, \dots, {\vec{v}}_{n}, \vec{0}}$ in a vector space $V$ to be linearly independent? Recall that $\vec{0}$ represents the additive identity.

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Subsection 2.5.2 Videos

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Figure 17. Video: Linear independence

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Exploration 2.5.15.

Prove the result of Observation 2.5.8, by showing that, given a set

S = {{\vec{v}}_{1}, {\vec{v}}_{2}, \dots, {\vec{v}}_{n}}

of vectors,

S

is linearly independent iff the equation

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

is only true when

x_{1} = x_{2} = \dots = x_{n} = 0 .

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Subsection 2.5.6 Sample Problem and Solution

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Sample problem Example B.1.9.

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