TBIL-LA Spanning Sets (VS3)

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Section 2.3 Spanning Sets (VS3)

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Learning Outcomes

Determine if a set of Euclidean vectors spans $R^{n}$ by solving appropriate vector equations.

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Subsection 2.3.1 Class Activities

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Observation 2.3.1.

Any single non-zero vector/number $x$ in $R^{1}$ spans $R^{1},$ since $R^{1} = {c x | c \in R} .$

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Activity 2.3.2.

How many vectors are required to span $R^{2} ?$ Sketch a drawing in the $x y$ plane to support your answer.

$1$
$2$
$3$
$4$
Infinitely Many

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Activity 2.3.3.

How many vectors are required to span $R^{3} ?$

$1$
$2$
$3$
$4$
Infinitely Many

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Fact 2.3.4.

At least $n$ vectors are required to span $R^{n} .$

Figure 11. Failed attempts to span $R^{n}$ by $< n$ vectors

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Activity 2.3.5.

Choose any vector $[\begin{matrix} ? \\ ? \\ ? \end{matrix}]$ in $R^{3}$ that is not in $span {[\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}], [\begin{matrix} - 2 \\ 0 \\ 1 \end{matrix}]}$ by using technology to verify that $RREF [\begin{array}{ccc} 1 & - 2 & ? \\ - 1 & 0 & ? \\ 0 & 1 & ? \end{array}] = [\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] .$ (Why does this work?)

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Fact 2.3.6.

The set ${{\vec{v}}_{1}, \dots, {\vec{v}}_{m}}$ fails to span all of $R^{n}$ exactly when the vector equation

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

is inconsistent for some vector $\vec{w} .$

Note that this happens exactly when $RREF [{\vec{v}}_{1} \dots {\vec{v}}_{m}]$ has a non-pivot row of zeros.

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

\Rightarrow [\begin{array}{ccc} 1 & - 2 & a \\ - 1 & 0 & b \\ 0 & 1 & c \end{array}] \sim [\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}] for some choice of vector [\begin{matrix} a \\ b \\ c \end{matrix}] .

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Activity 2.3.7.

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)

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

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(b)

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

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Activity 2.3.8.

Consider the set of third-degree polynomials

\begin{aligned} S = { & 2 x^{3} + 3 x^{2} - 1, 2 x^{3} + 3, 3 x^{3} + 13 x^{2} + 7 x + 16, \\ - x^{3} + 10 x^{2} + 7 x + 14, 4 x^{3} + 3 x^{2} + 2} . \end{aligned}

and the question “Does $P_{3} = span S ?$ ”

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(a)

Rewrite this question to be about the solutions to a polynomial equation.

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(b)

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

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Activity 2.3.9.

Consider the set of matrices

S = {[\begin{array}{cc} 1 & 3 \\ 0 & 1 \end{array}], [\begin{array}{cc} 1 & - 1 \\ 1 & 0 \end{array}], [\begin{array}{cc} 1 & 0 \\ 0 & 2 \end{array}]}

and the question “Does $M_{2, 2} = span S ?$ ”

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(a)

Rewrite this as a question about the solutions to a matrix equation.

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(b)

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

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Activity 2.3.10.

Let ${\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3} \in R^{7}$ be three 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 $span {{\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}} .$
$span {\vec{w}, {\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}} = span {{\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}} .$
$span {\vec{w}, {\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}}$ is smaller than $span {{\vec{v}}_{1}, {\vec{v}}_{2}, {\vec{v}}_{3}} .$

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Subsection 2.3.2 Videos

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Figure 12. Video: Determining if a set spans a Euclidean space

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Exploration 2.3.11.

Construct each of the following, or show that it is impossible:

A set of 2 vectors that spans $R^{3}$
A set of 3 vectors that spans $R^{3}$
A set of 3 vectors that does not span $R^{3}$
A set of 4 vectors that spans $R^{3}$

For any of the sets you constructed that did span the required space, are any of the vectors a linear combination of the others in your set?

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Exploration 2.3.12.

Based on these results, generalize this a conjecture about how a set of

n - 1, n

and

n + 1

vectors would or would not span

R^{n} .

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Subsection 2.3.6 Sample Problem and Solution

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Sample problem Example B.1.7.

You have attempted 1 of 1 activities on this page.

Section 2.3 Spanning Sets (VS3)

Learning Outcomes

Subsection 2.3.1 Class Activities

Observation 2.3.1.

Activity 2.3.2.

Activity 2.3.3.

Fact 2.3.4.

Activity 2.3.5.

Fact 2.3.6.

Activity 2.3.7.

(a)

(b)

Activity 2.3.8.

(a)

(b)

Activity 2.3.9.

(a)

(b)

Activity 2.3.10.

Subsection 2.3.2 Videos

Subsection 2.3.3 Slideshow

Exercises 2.3.4 Exercises

Subsection 2.3.5 Mathematical Writing Explorations

Exploration 2.3.11.

Exploration 2.3.12.

Subsection 2.3.6 Sample Problem and Solution