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

Name	Status	Author	Last Modified
Represent a linear map as a matrix given a 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 given a basis	draft	John Steele	13/05/2019 04:30
Simon's copy of Represent a linear map as a matrix given a basis	draft	Simon Thomas	12/06/2019 15:52

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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:

$\phi(p(x))=p(\var{a})+p(\simplify{{b}x+{c}})$

This is given by evaluating $p(x)$ at $x=\var{a}$ and adding this to the polynomial given by replacing $x$ by $\simplify{{b}x+{c}}$ in $p(x)$ .

For example:

$\phi(x^2+2x)=\simplify[all,!collectnumbers,!noleadingminus]{{a}^2+2*{a}+({b}x+{c})^2+2*({b}x+{c})={a^2+2*a+2*c}+{2*b*c+2*b}*x+{b^2}*x^2}$ .

Using the standard basis for range and domain find the matrix given by $\phi$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

a	integer	-2
c	integer	4
b	integer	1

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random(-5..5 except 0)

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Generated value: `integer`

-2

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

Advice

We have:

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

gives the first column of the matrix.

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

gives the second column of the matrix.

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

gives the third column of the matrix.

Continuing on in this way for $\phi(x^3),\;\phi(x^4)$ :

$\phi(x^3)=\simplify[]{ {a}^ 3 + ({b} * x + {c})^3}= ...$

$\phi(x^4)=\simplify[]{ {a}^ 4 + ({b} * x + {c})^4}= ...$

we obtain the matrix for $\phi$ with respect to the given bases for domain and range.

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

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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 given a basis

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Simon Thomas 5 years, 10 months ago

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Generated value: `integer`

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

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Generated value: integer

Parts

Gap-fill

Generated value: `integer`