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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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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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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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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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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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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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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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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

Lectures
8 Videos

`\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

*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

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Conceptual explanation of Spanning Sets

*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

Therefore, `\vec a, \vec b`

spans `\mathbb{R} ^2`

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How to prove if two vectors span 2D

*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

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

*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