Linear Combinations (V2)

Section 2.2 Linear Combinations (V2)

Definition 2.2.1.

A linear combination of a set of vectors ${{\vec{v}}_{1}, {\vec{v}}_{2}, \dots, {\vec{v}}_{m}}$ is given by $c_{1} {\vec{v}}_{1} + c_{2} {\vec{v}}_{2} + \dots + c_{m} {\vec{v}}_{m}$ for any choice of scalar multiples $c_{1}, c_{2}, \dots, c_{m} .$

For example, we can say $[\begin{matrix} 3 \\ 0 \\ 5 \end{matrix}]$ is a linear combination of the vectors $[\begin{matrix} 1 \\ - 1 \\ 2 \end{matrix}]$ and $[\begin{matrix} 1 \\ 2 \\ 1 \end{matrix}]$ since

[\begin{matrix} 3 \\ 0 \\ 5 \end{matrix}] = 2 [\begin{matrix} 1 \\ - 1 \\ 2 \end{matrix}] + 1 [\begin{matrix} 1 \\ 2 \\ 1 \end{matrix}] .

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Definition 2.2.2.

The span of a set of vectors is the collection of all linear combinations of that set:

span {{\vec{v}}_{1}, {\vec{v}}_{2}, \dots, {\vec{v}}_{m}} = {c_{1} {\vec{v}}_{1} + c_{2} {\vec{v}}_{2} + \dots + c_{m} {\vec{v}}_{m} | c_{i} \in R} .

For example:

span {[\begin{matrix} 1 \\ - 1 \\ 2 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 1 \end{matrix}]} = {a [\begin{matrix} 1 \\ - 1 \\ 2 \end{matrix}] + b [\begin{matrix} 1 \\ 2 \\ 1 \end{matrix}] | a, b \in R} .

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Activity 2.2.3.

Consider $span {[\begin{matrix} 1 \\ 2 \end{matrix}]} .$

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(a)

Sketch $1 [\begin{matrix} 1 \\ 2 \end{matrix}] = [\begin{matrix} 1 \\ 2 \end{matrix}],$ $3 [\begin{matrix} 1 \\ 2 \end{matrix}] = [\begin{matrix} 3 \\ 6 \end{matrix}],$ $0 [\begin{matrix} 1 \\ 2 \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}],$ and $- 2 [\begin{matrix} 1 \\ 2 \end{matrix}] = [\begin{matrix} - 2 \\ - 4 \end{matrix}]$ in the $x y$ plane.

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(b)

Sketch a representation of all the vectors belonging to $span {[\begin{matrix} 1 \\ 2 \end{matrix}]} = {a [\begin{matrix} 1 \\ 2 \end{matrix}] | a \in R}$ in the $x y$ plane.

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Activity 2.2.4.

Consider $span {[\begin{matrix} 1 \\ 2 \end{matrix}], [\begin{matrix} - 1 \\ 1 \end{matrix}]} .$

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(a)

Sketch the following linear combinations in the $x y$ plane.

1 [\begin{matrix} 1 \\ 2 \end{matrix}] + 0 [\begin{matrix} - 1 \\ 1 \end{matrix}] 0 [\begin{matrix} 1 \\ 2 \end{matrix}] + 1 [\begin{matrix} - 1 \\ 1 \end{matrix}] 1 [\begin{matrix} 1 \\ 2 \end{matrix}] + 1 [\begin{matrix} - 1 \\ 1 \end{matrix}]

- 2 [\begin{matrix} 1 \\ 2 \end{matrix}] + 1 [\begin{matrix} - 1 \\ 1 \end{matrix}] - 1 [\begin{matrix} 1 \\ 2 \end{matrix}] + - 2 [\begin{matrix} - 1 \\ 1 \end{matrix}]

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(b)

Sketch a representation of all the vectors belonging to $span {[\begin{matrix} 1 \\ 2 \end{matrix}], [\begin{matrix} - 1 \\ 1 \end{matrix}]} = {a [\begin{matrix} 1 \\ 2 \end{matrix}] + b [\begin{matrix} - 1 \\ 1 \end{matrix}] | a, b \in R}$ in the $x y$ plane.

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Activity 2.2.5.

Sketch a representation of all the vectors belonging to $span {[\begin{matrix} 6 \\ - 4 \end{matrix}], [\begin{matrix} - 3 \\ 2 \end{matrix}]}$ in the $x y$ plane.

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Activity 2.2.6.

The vector $[\begin{matrix} - 1 \\ - 6 \\ 1 \end{matrix}]$ belongs to $span {[\begin{matrix} 1 \\ 0 \\ - 3 \end{matrix}], [\begin{matrix} - 1 \\ - 3 \\ 2 \end{matrix}]}$ exactly when there exists a solution to the vector equation $x_{1} [\begin{matrix} 1 \\ 0 \\ - 3 \end{matrix}] + x_{2} [\begin{matrix} - 1 \\ - 3 \\ 2 \end{matrix}] = [\begin{matrix} - 1 \\ - 6 \\ 1 \end{matrix}] .$

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(a)

Reinterpret this vector equation as a system of linear equations.

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(b)

Find its solution set, using technology to find $RREF$ of its corresponding augmented matrix.

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(c)

Given this solution set, does $[\begin{matrix} - 1 \\ - 6 \\ 1 \end{matrix}]$ belong to $span {[\begin{matrix} 1 \\ 0 \\ - 3 \end{matrix}], [\begin{matrix} - 1 \\ - 3 \\ 2 \end{matrix}]} ?$

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Fact 2.2.7.

A vector $\vec{b}$ belongs to $span {{\vec{v}}_{1}, \dots, {\vec{v}}_{n}}$ if and only if the vector equation $x_{1} {\vec{v}}_{1} + \dots + x_{n} {\vec{v}}_{n} = \vec{b}$ is consistent.

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Observation 2.2.8.

The following are all equivalent statements:

The vector $\vec{b}$ belongs to $span {{\vec{v}}_{1}, \dots, {\vec{v}}_{n}} .$
The vector equation $x_{1} {\vec{v}}_{1} + \dots + x_{n} {\vec{v}}_{n} = \vec{b}$ is consistent.
The linear system corresponding to $[{\vec{v}}_{1} \dots {\vec{v}}_{n} | \vec{b}]$ is consistent.
$RREF [{\vec{v}}_{1} \dots {\vec{v}}_{n} | \vec{b}]$ doesn't have a row $[0 \dots 0 | 1]$ representing the contradiction $0 = 1 .$

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Activity 2.2.9.

Determine if $[\begin{matrix} 3 \\ - 2 \\ 1 \\ 5 \end{matrix}]$ belongs to $span {[\begin{matrix} 1 \\ 0 \\ - 3 \\ 2 \end{matrix}], [\begin{matrix} - 1 \\ - 3 \\ 2 \\ 2 \end{matrix}]}$ by solving an appropriate vector equation.

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Activity 2.2.10.

Determine if $[\begin{matrix} - 1 \\ - 9 \\ 0 \end{matrix}]$ belongs to $span {[\begin{matrix} 1 \\ 0 \\ - 3 \end{matrix}], [\begin{matrix} - 1 \\ - 3 \\ 2 \end{matrix}]}$ by solving an appropriate vector equation.

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Activity 2.2.11.

Does the third-degree polynomial $3 y^{3} - 2 y^{2} + y + 5$ in $P_{3}$ belong to $span {y^{3} - 3 y + 2, - y^{3} - 3 y^{2} + 2 y + 2} ?$

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(a)

Reinterpret this question as a question about the solution(s) of a polynomial equation.

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(b)

Answer this equivalent question, and use its solution to answer the original question.

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Activity 2.2.12.

Does the polynomial $x^{2} + x + 1$ belong to $span {x^{2} - x, x + 1, x^{2} - 1} ?$

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Activity 2.2.13.

Does the matrix $[\begin{array}{cc} 3 & - 2 \\ 1 & 5 \end{array}]$ belong to $span {[\begin{array}{cc} 1 & 0 \\ - 3 & 2 \end{array}], [\begin{array}{cc} - 1 & - 3 \\ 2 & 2 \end{array}]} ?$

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(a)

Reinterpret this question as a question about the solution(s) of a matrix equation.

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(b)

Answer this equivalent question, and use its solution to answer the original question.

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Subsection 2.2.1 Videos

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Figure 2.2.14. Video: Linear combinations

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Exercises 2.2.2 Exercises

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Exercises available at checkit.clontz.org ¹.

https://checkit.clontz.org/#/banks/tbil-la/V2/