Simon's copy of Represent a linear map as a matrix with a given basis

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Simon Thomas 5 years, 10 months ago

Created this as a copy of Represent a linear map as a matrix with a given basis.

Name	Status	Author	Last Modified
Represent a linear map as a matrix with a given basis	Ready to use	Newcastle University Mathematics and Statistics	20/11/2019 14:50
John's copy of Represent a linear map as a matrix with a given basis	draft	John Steele	13/05/2019 04:30
Simon's copy of Represent a linear map as a matrix with a given basis	draft	Simon Thomas	13/06/2019 09:10

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Question statement

Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$ with coefficients in the real numbers.

Let the linear map

$\phi: P_4 \rightarrow P_4$ be defined by:

$\phi(p(x)) = \simplify[all,!collectnumbers]{{a} * p(x) + ({b} * x + {c}) * p'(x) + (x ^ 2 + {d} * x + {f}) * p''(x)}$

where $p'(x)$ is the first derivative of $p(x)$ and $p''(x)$ the second derivative.

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	5
c	integer	-3
b	integer	4
d	integer	4
f	integer	6

Automatically regenerate variables?

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xxxxxxxxxx
 
random(1..9 except 0)

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Description

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

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Generated value: `integer`

→ Used by:

Statement
Advice
"Unnamed part" → "Gap 0." - Minimum accepted value
"Unnamed part" → "Gap 0." - Maximum accepted value
"Unnamed part" → "Gap 4." - Minimum accepted value
"Unnamed part" → "Gap 4." - Maximum accepted value
"Unnamed part" → "Gap 8." - Minimum accepted value
"Unnamed part" → "Gap 8." - Maximum accepted value
"Unnamed part" → "Gap 12." - Minimum accepted value
"Unnamed part" → "Gap 12." - Maximum accepted value
"Unnamed part" → "Gap 15." - Minimum accepted value
"Unnamed part" → "Gap 15." - Maximum accepted value

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Using the ordered basis

$\{1,\;x,\;x^2,\;x^3,\;x^4\}$ of

$P_4$ for both range and domain,

$\phi$ is represented by a 5 x 5 matrix. Fill in the entries for this matrix below:

$\left( \begin{matrix}\phantom{.}\\ \phantom{.}\\ \phantom{.}\\ \phantom{.}\\ \phantom{.}\\ \phantom{.}\\ \phantom{.}\\ \phantom{.}\\ \phantom{.}\\ \end{matrix} \right.$	Gap 0.	Gap 1.	Gap 2.	$0$	$0$	$\left) \begin{matrix}\phantom{.}\\ \phantom{.}\\ \phantom{.}\\ \phantom{.}\\ \phantom{.}\\ \phantom{.}\\ \phantom{.} \\ \phantom{.}\\ \phantom{.}\\\end{matrix} \right.$
	Gap 3.	Gap 4.	Gap 5.	Gap 6.	$0$
	$0$	Gap 7.	Gap 8.	Gap 9.	Gap 10.
	$0$	$0$	Gap 11.	Gap 12.	Gap 13.
	$0$	$0$	$0$	Gap 14.	Gap 15.

Advice

We have:

$\phi(1) =\simplify[]{ {a} * 1 + ({b} * x + {c}) * 0 + (x ^ 2 + {d} * x + {f}) * 0 = {a} = {a} * 1 + 0 * x + 0 * x ^ 2 + 0 * x ^ 3 + 0 * x ^ 4}$

which gives the first column of the matrix.

$\phi(x) = \simplify[]{{a} * x + ({b} * x + {c}) * 1 + (x ^ 2 + {d} * x + {f}) * 0 = {c} + {a + b} * x = {c} * 1 + {a + b} * x + 0 * x ^ 2 + 0 * x ^ 3 + 0 * x ^ 4}$

gives the second column of the matrix.

$\phi(x ^ 2) = \simplify[]{{a} * x ^ 2 + ({b} * x + {c}) * 2 * x + (x ^ 2 + {d} * x + {f}) * 2 = {2 * f} + {2 * d + 2 * c} * x + {a + 2 * b + 2} * x ^ 2 = {2 * f} * 1 + {2 * d + 2 * c} * x + {a + 2 * b + 2} * x ^ 2 + 0 * x ^ 3 + 0 * x ^ 4}$

gives the third column of the matrix.

Continuing on in this way for $\phi(x^3),\;\phi(x^4)$ we obtain the matrix for $\phi$ with respect to the given bases for domain and range.

$\begin{pmatrix}\var{a}&\var{c}&\var{2*f}&0&0\\0&\var{a+b}&\var{2*d+2*c}&\var{6*f}&0\\0&0&\var{a+2*b+2}&\var{3*c+6*d}&\var{12*f}\\0&0&0&\var{a+3*b+6}&\var{4*c+12*d}\\0&0&0&0&\var{a+4*b+12}\end{pmatrix}$

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This question is used in the following exams:

Linear algebra by Simon Thomas in Maths support.
more sophisticated mathematics examples by Josh Lim in NUMBAS workshop demo.

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Simon's copy of Represent a linear map as a matrix with a given basis

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Simon Thomas 5 years, 10 months ago

Error in variable testing condition

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Generated value: `integer`

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Simon's copy of Represent a linear map as a matrix with a given basis

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 5 years, 10 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Gap-fill

Generated value: `integer`