copy of General linear combinations of standard basis vectors

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Julia Goedecke

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4 years, 7 months ago

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Created this as a copy of General linear combinations of standard basis vectors.

Name	Status	Author	Last Modified
Vectors - Introductory exercise (WB Q1.1 randomised)	Ready to use	Julia Goedecke	25/08/2021 12:57
General linear combinations of standard basis vectors	Ready to use	Julia Goedecke	25/08/2021 12:57
Demo accept NA or any counter-example	Ready to use	Julia Goedecke	25/08/2021 12:57
copy of General linear combinations of standard basis vectors	Ready to use	Julia Goedecke	25/08/2021 12:57

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Question statement

This question is about general linear combinations.

Give any introductory information the student needs.

			Name	Type	Generated Value

v1	vector	vector(4,3)
v2	vector	vector(2,4)
w1	vector	vector(2.2360679775,pi)
w2	vector	vector(-1.1180339887,pi)
n	integer	5
b	integer	2
mu1	integer	3
mu2	integer	6
e1	string	\begin\{pmatrix\}1\\0\\0\\0\en
e2	string	\begin\{pmatrix\}0\\1\\0\\0\en
e3	string	\begin\{pmatrix\}0\\0\\1\\0\en
e4	string	\begin\{pmatrix\}0\\0\\0\\1\en

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
vector((repeat(random(1..5),2)))

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

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Generated value: `vector`

vector(4,3)

→ Used by:

Variable testing condition

Parts

Part a) Gap-fill
1. Gap Gap 0. Mathematical expression
  Add
2. Gap Gap 1. Mathematical expression
  Add
Part b) Gap-fill
Part c) Gap-fill

Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Can you write any vector $\begin{pmatrix}x\\y\end{pmatrix}\in \mathbb{R}^2$ as a linear combination of the vectors $\begin{pmatrix}1\\0\end{pmatrix},\ \begin{pmatrix}0\\1\end{pmatrix}$ ?

If yes, fill in the coefficients. If no, write NA into the gaps.

$\begin{pmatrix}x\\y\end{pmatrix}=\$ Gap 0. $\begin{pmatrix}1\\0\end{pmatrix} +$ Gap 1. $\begin{pmatrix}0\\1\end{pmatrix}$ .

Advice

Part a) Whatever real numbers the entries $x$ and $y$ are, we can use those numbers as coefficients. So in general, we have $\begin{pmatrix}x\\y\end{pmatrix}=x\begin{pmatrix}1\\0\end{pmatrix}+y\begin{pmatrix}0\\1\end{pmatrix}$ .

Part b) Similarly, we have $\begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix}=x_1\var{latex(e1)}+x_2\var{latex(e2)}+x_3\var{latex(e3)} + x_4\var{latex(e4)}$ . Therefore, there is no counterexample.

Part c) We cannot write any vector in $\mathbb{R}^4$ as a linear combination of just the first three vectors: any vector which has a non-zero fourth entry cannot be written like this, because

$x_1\var{latex(e1)}+x_2\var{latex(e2)}+x_3\var{latex(e3)} =\begin{pmatrix}x_1\\x_2\\x_3\\0\end{pmatrix}.$

Whatever we choose for $x_1,\ x_2,\ x_3$ , we can never get a non-zero entry in the fourth position. For example, $\begin{pmatrix}0\\0\\0\\1\end{pmatrix}$ is a counter-example. Any vector with non-zero fourth entry is accepted as a correct counter-example.

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Part	Test	Passed?
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copy of General linear combinations of standard basis vectors

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Taxonomy: mathcentre

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Julia Goedecke 4 years, 7 months ago

Julia Goedecke 4 years, 7 months ago

Julia Goedecke 4 years, 7 months ago

Error in variable testing condition

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Generated value: `vector`

Parts

Gap-fill

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copy of General linear combinations of standard basis vectors

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Julia Goedecke 4 years, 7 months ago

Julia Goedecke 4 years, 7 months ago

Julia Goedecke 4 years, 7 months ago

Error in variable testing condition

Error

Warning

Generated value: vector

Parts

Gap-fill

Generated value: `vector`