Mixing
Linear Algebra, how to solve transformation T: ℙ2→M 2,2?
$begingroup$
Could someone please help me with two questions?
enter image description here
enter image description here
So I know they must be a basis of P3, so
p = a + bx + cx^2 + dx^3
I then need to find a way to sub the equation inside but I have no idea how to do so.
I am using the Lyryx textbook, which did not really explain any of the steps so if anyone could teach me step by step, that would be great help!
Thank you!
linear-algebra linear-transformations
asked Jan 31 at 6:58
MichaelMichael
1
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add a comment |
$begingroup$
Could someone please help me with two questions?
enter image description here
enter image description here
So I know they must be a basis of P3, so
p = a + bx + cx^2 + dx^3
I then need to find a way to sub the equation inside but I have no idea how to do so.
I am using the Lyryx textbook, which did not really explain any of the steps so if anyone could teach me step by step, that would be great help!
Thank you!
linear-algebra linear-transformations
asked Jan 31 at 6:58
MichaelMichael
1
$endgroup$
$begingroup$
Remember a special property of linear transformations, namely that $L(aoverrightarrow{x} + boverrightarrow{y}) = aL(overrightarrow{x}) + bL(overrightarrow{x})$. If you can get a linear combination of polynomials to equal another polynomial, use the corresponding combination of the matrices to get the transformation.
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– Hyperion
Jan 31 at 7:07
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elcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
$endgroup$
– José Carlos Santos
Jan 31 at 7:12
add a comment |
$begingroup$
Could someone please help me with two questions?
enter image description here
enter image description here
So I know they must be a basis of P3, so
p = a + bx + cx^2 + dx^3
I then need to find a way to sub the equation inside but I have no idea how to do so.
I am using the Lyryx textbook, which did not really explain any of the steps so if anyone could teach me step by step, that would be great help!
Thank you!
linear-algebra linear-transformations
asked Jan 31 at 6:58
MichaelMichael
1
$endgroup$
Could someone please help me with two questions?
enter image description here
enter image description here
So I know they must be a basis of P3, so
p = a + bx + cx^2 + dx^3
I then need to find a way to sub the equation inside but I have no idea how to do so.
I am using the Lyryx textbook, which did not really explain any of the steps so if anyone could teach me step by step, that would be great help!
Thank you!
linear-algebra linear-transformations
linear-algebra linear-transformations
asked Jan 31 at 6:58
MichaelMichael
1
asked Jan 31 at 6:58
MichaelMichael
1
asked Jan 31 at 6:58
MichaelMichael
1
asked Jan 31 at 6:58
MichaelMichael
1
asked Jan 31 at 6:58
MichaelMichael
1
1
$begingroup$
Remember a special property of linear transformations, namely that $L(aoverrightarrow{x} + boverrightarrow{y}) = aL(overrightarrow{x}) + bL(overrightarrow{x})$. If you can get a linear combination of polynomials to equal another polynomial, use the corresponding combination of the matrices to get the transformation.
$endgroup$
– Hyperion
Jan 31 at 7:07
$begingroup$
elcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
$endgroup$
– José Carlos Santos
Jan 31 at 7:12
add a comment |
$begingroup$
Remember a special property of linear transformations, namely that $L(aoverrightarrow{x} + boverrightarrow{y}) = aL(overrightarrow{x}) + bL(overrightarrow{x})$. If you can get a linear combination of polynomials to equal another polynomial, use the corresponding combination of the matrices to get the transformation.
$endgroup$
– Hyperion
Jan 31 at 7:07
$begingroup$
elcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
$endgroup$
– José Carlos Santos
Jan 31 at 7:12
$begingroup$
Remember a special property of linear transformations, namely that $L(aoverrightarrow{x} + boverrightarrow{y}) = aL(overrightarrow{x}) + bL(overrightarrow{x})$. If you can get a linear combination of polynomials to equal another polynomial, use the corresponding combination of the matrices to get the transformation.
$endgroup$
– Hyperion
Jan 31 at 7:07
$begingroup$
Remember a special property of linear transformations, namely that $L(aoverrightarrow{x} + boverrightarrow{y}) = aL(overrightarrow{x}) + bL(overrightarrow{x})$. If you can get a linear combination of polynomials to equal another polynomial, use the corresponding combination of the matrices to get the transformation.
$endgroup$
– Hyperion
Jan 31 at 7:07
$begingroup$
elcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
$endgroup$
– José Carlos Santos
Jan 31 at 7:12
$begingroup$
elcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
$endgroup$
– José Carlos Santos
Jan 31 at 7:12
add a comment |
1 Answer
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oldest
votes
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First one: $2q-p=rimplies T(r)=T(2p-q)=2T(p)-T(q)=begin{pmatrix}5&11\-21&0end{pmatrix}$.
Second: You get a system of $4$ equations in $4$ unknowns. $ax^3+bx^2+cx+d=a_1x^3+a_2x^2+a_3(x+1)+a_4(x^3+x^2+x+2)$.
Multiply through and set the coefficients equal. Solve the system.
I get: $a_1=a+c-d\a_2=b+c-d\a_3=2c-d \a_4=-c+d$.
Now use the matrices given and the linear property. I get
$begin{pmatrix}9a+3b-3d&-6a+3b\-5a-b+2c-6d&4a-b-3c+5dend{pmatrix}$.
answered Jan 31 at 9:37
Chris CusterChris Custer
14.3k3827
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$begingroup$
Hello, what is the process you took to get to that?
$endgroup$
– Michael
Jan 31 at 14:10
$begingroup$
Honestly I just looked at it to get $r=2q-p$. In general, you get a system of equations.
$endgroup$
– Chris Custer
Jan 31 at 14:13
add a comment |
Your Answer
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1 Answer
1
active
oldest
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1 Answer
1
active
oldest
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active
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active
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$begingroup$
First one: $2q-p=rimplies T(r)=T(2p-q)=2T(p)-T(q)=begin{pmatrix}5&11\-21&0end{pmatrix}$.
Second: You get a system of $4$ equations in $4$ unknowns. $ax^3+bx^2+cx+d=a_1x^3+a_2x^2+a_3(x+1)+a_4(x^3+x^2+x+2)$.
Multiply through and set the coefficients equal. Solve the system.
I get: $a_1=a+c-d\a_2=b+c-d\a_3=2c-d \a_4=-c+d$.
Now use the matrices given and the linear property. I get
$begin{pmatrix}9a+3b-3d&-6a+3b\-5a-b+2c-6d&4a-b-3c+5dend{pmatrix}$.
answered Jan 31 at 9:37
Chris CusterChris Custer
14.3k3827
$endgroup$
$begingroup$
Hello, what is the process you took to get to that?
$endgroup$
– Michael
Jan 31 at 14:10
$begingroup$
Honestly I just looked at it to get $r=2q-p$. In general, you get a system of equations.
$endgroup$
– Chris Custer
Jan 31 at 14:13
add a comment |
$begingroup$
First one: $2q-p=rimplies T(r)=T(2p-q)=2T(p)-T(q)=begin{pmatrix}5&11\-21&0end{pmatrix}$.
Second: You get a system of $4$ equations in $4$ unknowns. $ax^3+bx^2+cx+d=a_1x^3+a_2x^2+a_3(x+1)+a_4(x^3+x^2+x+2)$.
Multiply through and set the coefficients equal. Solve the system.
I get: $a_1=a+c-d\a_2=b+c-d\a_3=2c-d \a_4=-c+d$.
Now use the matrices given and the linear property. I get
$begin{pmatrix}9a+3b-3d&-6a+3b\-5a-b+2c-6d&4a-b-3c+5dend{pmatrix}$.
answered Jan 31 at 9:37
Chris CusterChris Custer
14.3k3827
$endgroup$
$begingroup$
Hello, what is the process you took to get to that?
$endgroup$
– Michael
Jan 31 at 14:10
$begingroup$
Honestly I just looked at it to get $r=2q-p$. In general, you get a system of equations.
$endgroup$
– Chris Custer
Jan 31 at 14:13
add a comment |
$begingroup$
First one: $2q-p=rimplies T(r)=T(2p-q)=2T(p)-T(q)=begin{pmatrix}5&11\-21&0end{pmatrix}$.
Second: You get a system of $4$ equations in $4$ unknowns. $ax^3+bx^2+cx+d=a_1x^3+a_2x^2+a_3(x+1)+a_4(x^3+x^2+x+2)$.
Multiply through and set the coefficients equal. Solve the system.
I get: $a_1=a+c-d\a_2=b+c-d\a_3=2c-d \a_4=-c+d$.
Now use the matrices given and the linear property. I get
$begin{pmatrix}9a+3b-3d&-6a+3b\-5a-b+2c-6d&4a-b-3c+5dend{pmatrix}$.
answered Jan 31 at 9:37
Chris CusterChris Custer
14.3k3827
$endgroup$
First one: $2q-p=rimplies T(r)=T(2p-q)=2T(p)-T(q)=begin{pmatrix}5&11\-21&0end{pmatrix}$.
Second: You get a system of $4$ equations in $4$ unknowns. $ax^3+bx^2+cx+d=a_1x^3+a_2x^2+a_3(x+1)+a_4(x^3+x^2+x+2)$.
Multiply through and set the coefficients equal. Solve the system.
I get: $a_1=a+c-d\a_2=b+c-d\a_3=2c-d \a_4=-c+d$.
Now use the matrices given and the linear property. I get
$begin{pmatrix}9a+3b-3d&-6a+3b\-5a-b+2c-6d&4a-b-3c+5dend{pmatrix}$.
answered Jan 31 at 9:37
Chris CusterChris Custer
14.3k3827
edited Jan 31 at 14:58
answered Jan 31 at 9:37
Chris CusterChris Custer
14.3k3827
answered Jan 31 at 9:37
Chris CusterChris Custer
14.3k3827
answered Jan 31 at 9:37
Chris CusterChris Custer
14.3k3827
14.3k3827
$begingroup$
Hello, what is the process you took to get to that?
$endgroup$
– Michael
Jan 31 at 14:10
$begingroup$
Honestly I just looked at it to get $r=2q-p$. In general, you get a system of equations.
$endgroup$
– Chris Custer
Jan 31 at 14:13
add a comment |
$begingroup$
Hello, what is the process you took to get to that?
$endgroup$
– Michael
Jan 31 at 14:10
$begingroup$
Honestly I just looked at it to get $r=2q-p$. In general, you get a system of equations.
$endgroup$
– Chris Custer
Jan 31 at 14:13
$begingroup$
Hello, what is the process you took to get to that?
$endgroup$
– Michael
Jan 31 at 14:10
$begingroup$
Hello, what is the process you took to get to that?
$endgroup$
– Michael
Jan 31 at 14:10
$begingroup$
Honestly I just looked at it to get $r=2q-p$. In general, you get a system of equations.
$endgroup$
– Chris Custer
Jan 31 at 14:13
$begingroup$
Honestly I just looked at it to get $r=2q-p$. In general, you get a system of equations.
$endgroup$
– Chris Custer
Jan 31 at 14:13
add a comment |
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$begingroup$
Remember a special property of linear transformations, namely that $L(aoverrightarrow{x} + boverrightarrow{y}) = aL(overrightarrow{x}) + bL(overrightarrow{x})$. If you can get a linear combination of polynomials to equal another polynomial, use the corresponding combination of the matrices to get the transformation.
$endgroup$
– Hyperion
Jan 31 at 7:07
$begingroup$
elcome to MSE. It is in your best interest that you type your questions (using MathJax) instead of posting links to pictures.
$endgroup$
– José Carlos Santos
Jan 31 at 7:12