Vectors and linear combinations

Section 2.1 Vectors and linear combinations

Activity 2.1.0.1.

Suppose that

v = [\begin{array}{r} 3 \\ 1 \end{array}], w = [\begin{array}{r} - 1 \\ 2 \end{array}] .

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(a)

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Find expressions for the vectors

\begin{array}{cccc} v, & 2 v, & - v, & - 2 v, \\ w, & 2 w, & - w, & - 2 w . \end{array}

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and sketch them in the

x y

plane.

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(b)

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What geometric effect does scalar multiplication by a positive number have on a vector? (A) preserves the length, changes the direction, (B) preserves the direction, changes the length, (C) changes the length and the direction.

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(c)

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What geometric effect does scalar multiplication by a negative number have on a vector? (A) preserves the length, changes the direction, (B) preserves the direction, changes the length, (C) changes the length and the direction.

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Activity 2.1.0.2.

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As in the previous activity, suppose that

v = [\begin{array}{r} 3 \\ 1 \end{array}], w = [\begin{array}{r} - 1 \\ 2 \end{array}] .

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(a)

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Sketch the vectors

v, w, v + w

in the

x y

plane.

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(b)

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Consider vectors that have the form

v + a w

where

a

is any scalar. Sketch a few of these vectors when, say,

a = - 2, - 1, 0, 1,

and

2 .

Consider the points at the tips of each of these vectors. An appropriate geometric description for this set of points would be

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a line through the tip of $v$ parallel to $w .$
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a line through the tip of $w$ parallel to $v .$
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a line through $(0, 0)$ parallel to $v + w .$

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Activity 2.1.0.3.

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As in the previous activity, suppose that

v = [\begin{array}{r} 3 \\ 1 \end{array}], w = [\begin{array}{r} - 1 \\ 2 \end{array}] .

🔗

(a)

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a

and

b

are two scalars, then the vector

a v + b w

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is called a linear combination of the vectors

v

and

w .

Find the vector that is the linear combination when

a = - 2

and

b = 1 .

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(b)

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Can the vector

[\begin{array}{r} - 31 \\ 37 \end{array}]

be represented as a linear combination of

v

and

w ?

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Activity 2.1.0.4.

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In this activity, we will look at linear combinations of a pair of vectors,

v = [\begin{array}{r} 2 \\ 1 \end{array}], w = [\begin{array}{r} 1 \\ 2 \end{array}]

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with weights

a

and

b .

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Go to the Applet in Activity 2.1.2 ² in Understanding Linear Algebra.

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(a)

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The weight

b

is initially set to 0. Explain what happens as you vary

a

with

b = 0 ?

How is this related to scalar multiplication?

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(b)

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What is the linear combination of

v

and

w

when

a = 1

and

b = - 2 ?

You may find this result using the diagram, but you should also verify it by computing the linear combination.

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(c)

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The vectors that arise when the weight

b

is set to 1 and

a

is varied is

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a line through $v$ parallel to $w .$
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a line through $w$ parallel to $v .$
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a line through $(0, 0)$ parallel to $v + w .$

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(d)

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Can the vector

[\begin{array}{r} 0 \\ 0 \end{array}]

be expressed as a linear combination of

v

and

w ?

If so, what are weights

a

and

b ?

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(e)

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Can the vector

[\begin{array}{r} 3 \\ 0 \end{array}]

be expressed as a linear combination of

v

and

w ?

If so, what are weights

a

and

b ?

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(f)

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Verify the result from the previous part by algebraically finding the weights

a

and

b

that form the linear combination

[\begin{array}{r} 3 \\ 0 \end{array}] .

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(g)

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Can the vector

[\begin{array}{r} 1.3 \\ - 1.7 \end{array}]

be expressed as a linear combination of

v

and

w ?

What about the vector

[\begin{array}{r} 15.2 \\ 7.1 \end{array}] ?

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(h)

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Are there any two-dimensional vectors that cannot be expressed as linear combinations of

v

and

w ?

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Activity 2.1.0.5.

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Given the vectors

v_{1} = [\begin{array}{r} 4 \\ 0 \\ 2 \\ 1 \end{array}], v_{2} = [\begin{array}{r} 1 \\ - 3 \\ 3 \\ 1 \end{array}], v_{3} = [\begin{array}{r} - 2 \\ 1 \\ 1 \\ 0 \end{array}], b = [\begin{array}{r} 0 \\ 1 \\ 2 \\ - 2 \end{array}],

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we ask if

b

can be expressed as a linear combination of

v_{1},

v_{2},

and

v_{3} .

Rephrase this question by writing a linear system for the weights

c_{1},

c_{2},

and

c_{3}

and use the Sage cell below to answer this question.

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Activity 2.1.0.6.

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Consider the following linear system.

\begin{aligned} 3 x_{1} & + & 2 x_{2} & - x_{3} & = & 4 \\ x_{1} & + 2 x_{3} & = & 0 \\ - x_{1} & - & x_{2} & + 3 x_{3} & = & 1 \end{aligned}

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Identify vectors

v_{1},

v_{2},

v_{3},

and

b

and rephrase the question "Is this linear system consistent?" by asking "Can

b

be expressed as a linear combination of

v_{1},

v_{2},

and

v_{3} ?

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Activity 2.1.0.7.

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Consider the vectors

v_{1} = [\begin{array}{r} 0 \\ - 2 \\ 1 \end{array}], v_{2} = [\begin{array}{r} 1 \\ 1 \\ - 1 \end{array}], v_{3} = [\begin{array}{r} 2 \\ 0 \\ - 1 \end{array}], b = [\begin{array}{r} - 1 \\ 3 \\ - 1 \end{array}] .

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(a)

🔗

Which statement is the most accurate for

b ?

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The vector $b$ can be expressed as a linear combination of $v_{1},$ $v_{2},$ and $v_{3}$ in exactly one way.
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The vector $b$ can be expressed as a linear combination of $v_{1},$ $v_{2},$ and $v_{3}$ in more than one way.
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The vector $b$ cannot be expressed as a linear combination of $v_{1},$ $v_{2},$ and $v_{3} .$

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🔗

(b)

🔗

Considering just the vectors

v_{1},

v_{2},

and

v_{3},

determine the most accurate statement.

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Every three dimensional vector $b$ can be expressed as a linear combination of $v_{1},$ $v_{2},$ and $v_{3}$ in exactly one way.
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Every three dimensional vector $b$ can be expressed as a linear combination of $v_{1},$ $v_{2},$ and $v_{3}$ in more than one way.
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There are vectors $b$ that cannot be expressed as a linear combination of $v_{1},$ $v_{2},$ and $v_{3} .$

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Be able to explain how the pivot positions of the matrix

[\begin{array}{rrr} v_{1} & v_{2} & v_{3} \end{array}]

help answer this question.

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Activity 2.1.0.8.

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Consider the vectors

v_{1} = [\begin{array}{r} 0 \\ - 2 \\ 1 \end{array}], v_{2} = [\begin{array}{r} 1 \\ 1 \\ - 1 \end{array}], v_{3} = [\begin{array}{r} 1 \\ - 1 \\ - 2 \end{array}], b = [\begin{array}{r} 0 \\ 8 \\ - 4 \end{array}] .

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(a)

🔗

Which statement is the most accurate for

b ?

🔗
The vector $b$ can be expressed as a linear combination of $v_{1},$ $v_{2},$ and $v_{3}$ in exactly one way.
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The vector $b$ can be expressed as a linear combination of $v_{1},$ $v_{2},$ and $v_{3}$ in more than one way.
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The vector $b$ cannot be expressed as a linear combination of $v_{1},$ $v_{2},$ and $v_{3} .$

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(b)

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Considering just the vectors

v_{1},

v_{2},

and

v_{3}

from the previous part, determine the most accurate statement.

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Every three dimensional vector $b$ can be expressed as a linear combination of $v_{1},$ $v_{2},$ and $v_{3}$ in exactly one way.
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Every three dimensional vector $b$ can be expressed as a linear combination of $v_{1},$ $v_{2},$ and $v_{3}$ in more than one way.
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There are vectors $b$ that cannot be expressed as a linear combination of $v_{1},$ $v_{2},$ and $v_{3} .$

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Be able to explain how the pivot positions of the matrix

[\begin{array}{rrr} v_{1} & v_{2} & v_{3} \end{array}]

help answer this question.

davidaustinm.github.io/ula/sec-vectors-lin-combs.html#activity-11

TBIL Activities for Understanding Linear Algebra

Search Results:

Section 2.1 Vectors and linear combinations

Activity 2.1.0.1.

(a)

(b)

(c)

Activity 2.1.0.2.

(a)

(b)

Activity 2.1.0.3.

(a)

(b)

Activity 2.1.0.4.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Activity 2.1.0.5.

Activity 2.1.0.6.

Activity 2.1.0.7.

(a)

(b)

Activity 2.1.0.8.

(a)

(b)