6.8 Linear Combination and Spanning Sets
Chapter
Chapter 6
Section
6.8
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Lectures 8 Videos

Introduction to Linear Combination

\vec{c} = k\vec{a} + l\vec{b}

That means \vec a, \vec b, \vec c are coplaner

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Introduction to Linear Combination

Geometrically example of linear combination of vectors ex

ex Three vectors \vec u, \vec v, and \vec x have magnitudes |\vec u| = 10, |\vec v| = 15, and |\vec x| = 24. If x lies between \vec u and \vec v in the same plane, making an angle of 20 with \vec u and 30 with \vec v, express \vec x as a linear combination of u and v.

Therefore, \vec x = 1.57 \vec u + 0.71 \vec v

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Geometrically example of linear combination of vectorsvector ex

Conceptual explanation of Spanning Sets

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Conceptual explanation of Spanning Sets

Finding coefficients to represent a vector in 2D as linear combination of two other given vectors

ex \vec x = (4, 36) as linear combination of

a) {(-1, 2), (2, 1)}

b) {(1, 0), (-2, 1)}

a) Therefore, \vec x = \dfrac{68}{5}(-1, 2) + \dfrac{44}{5}(2, 1)

b) Therefore, \vec x = 76(1, 0) + 36(-2, 1)

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Finding coefficients to represent a vector in 2D as linear combination of two other given vectors

How to prove if two vectors span 2D

Therefore, \vec a, \vec b spans \mathbb{R} ^2

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How to prove if two vectors span 2D

Why two parallel vector does not span vectors in 2D with an example

ex Span \vec a = (1, 3), \vec b = (2, 6)

Therefore, there is no solution.

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Why two parallel vector does not span vectors in 2D with an example

Conceptual explanation of why we need 3 non coplanar vectors to span 3D

3 linearly independent vectors will span \mathbb{R}^3.

\vec a, \vec b, \vec c

\vec a = m\vec b + n\vec c *

No m, n can be found so that * is true.

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Conceptual explanation of why we need 3 non coplanar vectors to span 3D

Selecting a Linear combination Strategy to determine if vectors lie on the same plane

ex

a) Given the two vectors \vec a = (1, 2, 1) and \vec b = (3, 1, 1), does the vector \vec c = (-9, 3, 1) lie on the plane determine by \vec a and \vec b? Explain.

Therefore, the vector \vec c does not lie on the same plane determined by \vec a and \vec b.

b) Does the vector (-1, 3, 1) lie in the plane determined by the first two vectors

Therefore, (-1, 3, 1) is coplanar with \vec a, \vec b

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Selecting a Linear combination Strategy to determine if vectors lie on the same plane
Solutions 19 Videos

A student writes 2(1, 0) + 4(-1, 0) =(-2,0( so (1, 0), (-1, 0) span R^2. What is wrong with this conclusion?

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Q1

It is claimed that \{(1, 0, 0), (0, 1, 0), (0, 0, 0)\} is a set of vectors spanning  R^3. Explain why it is not possible for these vectors to span R^3.

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Q2

Describe the set of vectors spanned by (0, 1). Say why this is the same set as that spanned by (0, -1).

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Q3

In R^3, the vector \vec{i} = (1, 0, 0), spans a set. Describe the set spanned by this vector. Name two other vectors that would also span the same set.

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Q4

It is proposed that the set \{(0, 0), (1, 0)\} could be used to span R^2. Explain why this is not possible.

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Q5

The following is a spanning set for R^2.

\{(-1, 2), (2, -4), (-1, 1), (-3, 6), (1, 0)\}

Remove three of the vectors and write down a spanning set that can be used to span R^2.

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Q6

Simplify each of the following linear combinations and write your answer in component form:

\vec{a} = \vec{i} - 2\vec{j}, \vec{b} = \vec{j} - 3\vec{k}, and \vec{c}= \vec{i} - 3\vec{j} + 2\vec{k}

2(2\vec{a} - 3\vec{b} + \vec{c}) - 4(-\vec{a} + \vec{b} - \vec{c}) + (\vec{a} -\vec{c})

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Q7a

Simplify each of the following linear combinations and write your answer in component form:

\vec{a} = \vec{i} - 2\vec{j}, \vec{b} = \vec{j} - 3\vec{k}, and \vec{c}= \vec{i} - 3\vec{j} + 2\vec{k}

\displaystyle{\frac{1}{2}}(2\vec{a} - 4\vec{b} - 8\vec{c}) - \displaystyle{\frac{1}{3}}(3\vec{a} - 6\vec{b} + 9\vec{c})

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Q7b

Name two sets of vectors that could be used to span the xy-plane in R^3. Show how the vectors (-1, 2, 0) and (3, 4, 0) could each be written as a linear combination of the vectors you have chosen.

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Q8

a) The set of vectors {(1, 0, 0), (0, 1, 0)} spans a set in \mathbb{R}^3. Describe this set.

b) Write the vector (-2, 4, 0) as a linear combination of these vectors.

c) Explain why it is not possible to write (3, 5, 8) as a linear combination of these vectors.

d) If the vector (1, 1, 0) were added to this set, what would these three vectors span in \mathbb{R}^3?

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Q9

Solve for a, b, and c in the following equation:

2(a, 3, c) + 3(c, 7, c) = (5, b + c, 15)

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Q10

Write the vector (-10, -34) as a linear combination of the vectors (-1, 3) and (1, 5).

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Q11

Find a and b which will satisfy to give you equation of a line give two position vectors \vec{u} and \vec{b}: (x, y) = a\vec{u} + b\vec{v}.

Given two position vectors {(2, -1), (-1, 1)}

Write each of the following vectors as a linear combination of the set given in part a: (2, -3), (124, -5), and (4, -11).

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Q12

Show that the vectors (-1,2, 3), (4, 1, -2), and (-14, -1, 16) do not lie on the same plane.

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Q13a

Show that the vectors (-1, 3, 4), (0, -1, 1), and (-3, 14, 7) lie on the same plane, and show how one of the vectors can be written as a linear combination of the other two.

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Q13b

Determine the value for x such that the points A(-1, 3, 4), B(-2, 3, -1), and C(-5, 6, x) all lie on a plane that contains the origin.

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Q14

The vectors \vec{a} and \vec{b} span \mathbb{R}^2. What values of m and n will make the following statements true: (m - 2)\vec{a} = (n + 3)\vec{b}? Explain your reasoning.

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Q15

The vectors (4, 1, 7), (-1, 1, 6) and (p, q, 5) are coplanar. Determine three sets of values for p and q for which this is true?

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Q16

The vectors \vec{a} and \vec{b} span \mathbb{R}^2. For what values of m is it true that (m^2 + 2m - 3)\vec{a} + (m^2 + m - 6)\vec{b} = \vec{0}? Explain your reasoning.

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Q17