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

Geometrically example of linear combination of vectorsvector ex

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7.46mins
Geometrically example of linear combination of vectorsvector ex

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

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3.53mins
Finding coefficients to represent a vector in 2D as linear combination of two other given vectors

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

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2.27mins
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

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3.42mins
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

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4.50mins
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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0.33mins
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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1.04mins
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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0.48mins
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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1.08mins
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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0.21mins
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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0.57mins
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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2.25mins
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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1.12mins
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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1.12mins
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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2.00mins
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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1.29mins
Q10

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

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2.16mins
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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2.50mins
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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3.47mins
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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1.40mins
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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2.52mins
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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0.55mins
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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3.53mins
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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2.17mins
Q17