Matrix multiplication via linear combinations of columns

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Julia Goedecke

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Julia Goedecke

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Question statement

Let $A=\var{A}$ and $B=\var{B}$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

A	matrix	Matrix of size 3×3
B	matrix	Matrix of size 3×3
A1	vector	vector(3,6,0)
A2	vector	vector(-2,5,4)
A3	vector	vector(7,4,9)
B1	vector	vector(6,0,7)
B2	vector	vector(-2,1,7)
B3	vector	vector(4,3,5)

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
matrix([3,-2,7],[6,5,4],[0,4,9])

JME function reference

Description

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

Can an exam override the value of this variable?

Generated value: `matrix`

matrix([3,-2,7],[6,5,4],[0,4,9])

→ Used by:

Statement
Advice
"Part a)" → "A^2" - Correct answer
"Part b)" → "Unnamed gap" - Correct answer

Parts

Part a) Gap-fill
1. Gap A^2 Matrix entry
  Add
2. Gap B^2 Matrix entry
  Add
Part b) Gap-fill
1. Gap Mathematical expression
  1. Alt Unnamed alternative
  Add
Part c) Gap-fill
1. Gap Mathematical expression
  1. Alt Unnamed alternative
  Add

Gap-fill

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

Compute

$A^2=$ A^2

$B^2=$ B^2

Advice

To answer b) and c), we use the definition of matrix multiplication which is in terms of the column vectors of the second matrix:

$B=\begin{pmatrix} \uparrow & &\uparrow & & \uparrow\\ b_1 & \cdots & b_k & \cdots & b_n\\ \downarrow & & \downarrow & & \downarrow \end{pmatrix},$
then

$AB=\begin{pmatrix} \uparrow & &\uparrow & & \uparrow\\ Ab_1 & \cdots & Ab_k & \cdots & Ab_n\\ \downarrow & & \downarrow & & \downarrow \end{pmatrix},$

and

$\begin{pmatrix} \phantom{.} & & & & \\ \uparrow & &\uparrow & & \uparrow\\ a_1 & \cdots & a_k & \cdots & a_n\\ \downarrow & & \downarrow & & \downarrow\\ \phantom{.}&&&& \end{pmatrix}\begin{pmatrix} x_1\\x_2\\\vdots\\x_{n-1}\\x_n\end{pmatrix} = x_1\begin{pmatrix} \\\uparrow\\a_1\\\downarrow\\ \ \end{pmatrix}+x_2\begin{pmatrix} \\\uparrow\\a_2\\\downarrow\\\ \end{pmatrix}+\cdots+x_{n-1}\begin{pmatrix} \\\uparrow\\a_{n-1}\\\downarrow\\\ \end{pmatrix}+x_n\begin{pmatrix} \\\uparrow\\a_n\\\downarrow\\\ \end{pmatrix}$

So the first column of $A^2$ is $(A^2)_1= Aa_1=A_{11}a_1+A_{21}a_2+A_{31}a_3=\var{A[0][0]}\var{A1}+\var{A[1][0]}\var{A2}+\var{A[2][0]}\var{A3}$ .

And the second column of $B^2$ is $(B^2)_2=Bb_2=B_{12}b_1+B_{22}b_2+B_{32}b_3=\var{B[0][1]}\var{b1}+\var{B[1][1]}\var{b2}+\var{B[2][1]}\var{b3}$ .

You can calculate these linear combinations to check your own answer.

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Part	Test	Passed?
Gap-fill
Gap-fill		Hasn't run yet
Matrix entry
Matrix entry		Hasn't run yet
Matrix entry
Matrix entry		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet
Gap-fill
Gap-fill		Hasn't run yet
Mathematical expression
Mathematical expression		Hasn't run yet

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Matrix multiplication via linear combinations of columns

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Checkpoint description

Julia Goedecke 4 years, 6 months ago

Julia Goedecke 4 years, 6 months ago

Julia Goedecke 4 years, 6 months ago

Error in variable testing condition

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Warning

Generated value: `matrix`

Parts

Gap-fill

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Error

Matrix multiplication via linear combinations of columns

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Julia Goedecke 4 years, 6 months ago

Julia Goedecke 4 years, 6 months ago

Julia Goedecke 4 years, 6 months ago

Error in variable testing condition

Error

Warning

Generated value: matrix

Parts

Gap-fill

Generated value: `matrix`