TBIL-LA Spanning Sets (EV2)

We’re guaranteed at least one solution if the RREF of the corresponding augmented matrix has no contradictions; likewise, we have no solutions if the RREF corresponds to the contradiction

0 = 1 .

Given

[\begin{array}{cccc} 1 & - 2 & - 2 & ? \\ - 1 & 0 & - 2 & ? \\ 0 & 1 & 2 & ? \end{array}] \sim [\begin{array}{cccc} 1 & 0 & 2 & ? \\ 0 & 1 & 2 & ? \\ 0 & 0 & 0 & ? \end{array}]

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we may conclude that the set does not span all of

R^{3}

because...

the row $[0 1 2 | ?]$ prevents a contradiction.
the row $[0 1 2 | ?]$ allows a contradiction.
the row $[0 0 0 | ?]$ prevents a contradiction.
the row $[0 0 0 | ?]$ allows a contradiction.

Answer.

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Fact 2.2.8.

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The set

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

spans all of

R^{n}

exactly when the vector equation

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

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is consistent for every vector

\vec{w} .

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Likewise, the set

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

fails to span all of

R^{n}

exactly when the vector equation

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

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is inconsistent for some vector

\vec{w} .

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Note these two possibilities are decided based on whether or not the RREF of the vector equation’s coefficient matrix (that is,

RREF [{\vec{v}}_{1} \dots {\vec{v}}_{n}]

) has either all pivot rows, or at least one non-pivot row (a row of zeroes):

[\begin{array}{cccc} 1 & - 2 & - 2 \\ - 1 & 0 & - 2 \\ 0 & 1 & 2 \end{array}] \sim [\begin{array}{cccc} 1 & 0 & 2 \\ 0 & 1 & 2 \\ 0 & 0 & 0 \end{array}] .

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Activity 2.2.9.

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Consider the set of vectors

S = {[\begin{matrix} 2 \\ 3 \\ 0 \\ - 1 \end{matrix}], [\begin{matrix} 1 \\ - 4 \\ 3 \\ 0 \end{matrix}], [\begin{matrix} 1 \\ 7 \\ - 3 \\ - 1 \end{matrix}], [\begin{matrix} 0 \\ 3 \\ 5 \\ 7 \end{matrix}], [\begin{matrix} 3 \\ 13 \\ 7 \\ 16 \end{matrix}]}

and the question “Does

R^{4} = span S ?

”

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(a)

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Rewrite this question in terms of the solutions to a vector equation.

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(b)

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Answer your new question, and use this to answer the original question.

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Activity 2.2.10.

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Let

{\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3} \in R^{7}

be three Euclidean vectors, and suppose

\vec{w}

is another vector with

\vec{w} \in span {{\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}} .

What can you conclude about

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

$span {\vec{w}, {\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}}$ is larger than $span {{\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}} .$
$span {\vec{w}, {\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}}$ is the same as $span {{\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}} .$
$span {\vec{w}, {\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}}$ is smaller than $span {{\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}} .$

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Subsection 2.2.3 Individual Practice

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Activity 2.2.11.

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One of our important results in this lesson is Fact 2.2.5, which states that a set of

n

vectors is required to span

R^{n} .

While we developed some geometric intuition for why this true, we did not prove it in class. Before coming to class next time, follow the steps outlined below to convince yourself of this fact using the concepts we learned in this lesson.

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