Computing the adjoint of a linear operator Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Inner Product and Orthogonal PolynomialsWhat are common notations for the endomorphism group of a vector space?Find an ordered basis $B$ for $M_ntimes n(mathbbR)$ such that $[T]B$ is a diagonal matrix for $n > 2$Why is $dim(W)=3$?Adjoint Operator and InverseLinear transformation is self-adjoint iff its associated matrix in the standard basis is symmetricWhat does adjoint of a linear map?Linear Algebra TextbookLinear operator and orthonormal basisSimultaneous diagonalization of self adjoint matrices
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Computing the adjoint of a linear operator
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Inner Product and Orthogonal PolynomialsWhat are common notations for the endomorphism group of a vector space?Find an ordered basis $B$ for $M_ntimes n(mathbbR)$ such that $[T]B$ is a diagonal matrix for $n > 2$Why is $dim(W)=3$?Adjoint Operator and InverseLinear transformation is self-adjoint iff its associated matrix in the standard basis is symmetricWhat does adjoint of a linear map?Linear Algebra TextbookLinear operator and orthonormal basisSimultaneous diagonalization of self adjoint matrices
$begingroup$
I am taking an linear algebra class and we are on inner product space. I have a question that wants us to evaluate $T^*$ at a given vector. This is not a homework question, it is a textbook question, specifically question 3c in 6.3 of Linear Algebra by Friedberg, Insel, Spence (pg 366). I will it write out though:
$V=P_1(R)$ (the set of polynomials with degree $leq 1$ with real coefficients) with $langle f,grangle=int_-1^1f(t)g(t)dt,$ $T(f)=f'+3f.$ Compute $T^*(f(t))$ where $f(t)=4-2t$.
I know the answer (it's in the textbook) but I am not sure how to obtain the solution. It's my understanding that since $V$ is finite-dimensional we know there is an adjoint and that for all $f,gin P_1(R)$ $langle T(g),frangle=langle g,T^*(f)rangle$. I tried setting $f=4-2t$ and I thought maybe set $g=1$ to make my computation easier but I don't think I am getting this right. I also decided to take the standard basis for $P_1(R)$ and used Gram-Schmidt to obtain an orthonormal basis $gamma = frac1sqrt2,sqrtfrac32x$. I am not really sure what to do next, like do I need to calculate the exact adjoint map or something. Any help would be greatly appreciated.
UPDATE: I figured out based on the comment made by darij grinberg. I used the fact that $langle T(g),frangle=langle g,T^*(f)rangle$ where $g=1$ and $g=t$ and setting $T^*(f) = a+bt$ and figuring out $a$ and $b$ and then using that to compute $T^*(4-2t)$.
linear-algebra linear-transformations
asked Apr 2 at 0:56
InsigMathInsigMath
1,01021021
$endgroup$
|
show 1 more comment
$begingroup$
I am taking an linear algebra class and we are on inner product space. I have a question that wants us to evaluate $T^*$ at a given vector. This is not a homework question, it is a textbook question, specifically question 3c in 6.3 of Linear Algebra by Friedberg, Insel, Spence (pg 366). I will it write out though:
$V=P_1(R)$ (the set of polynomials with degree $leq 1$ with real coefficients) with $langle f,grangle=int_-1^1f(t)g(t)dt,$ $T(f)=f'+3f.$ Compute $T^*(f(t))$ where $f(t)=4-2t$.
I know the answer (it's in the textbook) but I am not sure how to obtain the solution. It's my understanding that since $V$ is finite-dimensional we know there is an adjoint and that for all $f,gin P_1(R)$ $langle T(g),frangle=langle g,T^*(f)rangle$. I tried setting $f=4-2t$ and I thought maybe set $g=1$ to make my computation easier but I don't think I am getting this right. I also decided to take the standard basis for $P_1(R)$ and used Gram-Schmidt to obtain an orthonormal basis $gamma = frac1sqrt2,sqrtfrac32x$. I am not really sure what to do next, like do I need to calculate the exact adjoint map or something. Any help would be greatly appreciated.
UPDATE: I figured out based on the comment made by darij grinberg. I used the fact that $langle T(g),frangle=langle g,T^*(f)rangle$ where $g=1$ and $g=t$ and setting $T^*(f) = a+bt$ and figuring out $a$ and $b$ and then using that to compute $T^*(4-2t)$.
linear-algebra linear-transformations
asked Apr 2 at 0:56
InsigMathInsigMath
1,01021021
$endgroup$
$begingroup$
What does $P_1left(Rright)$ mean? Polynomials of degree $leq 1$ with real coefficients?
$endgroup$
– darij grinberg
Apr 2 at 1:00
$begingroup$
Oh yes, sorry exactly @darijgrinberg
$endgroup$
– InsigMath
Apr 2 at 1:01
2
$begingroup$
The simplest way to compute $T^*(f)$ is probably to set $T^*(f) = a + bt$ with unknown constants $a$ and $b$, and then solve the system of the two linear equations $left<T(g), fright> = left<g, T^*(f)right>$ for $g = 1$ and $g = t$. (You have to rewrite these equations as explicit linear equations in $a$ and $b$, but you can do this, since the bilinear form is given as an easily computable integral.)
$endgroup$
– darij grinberg
Apr 2 at 1:04
$begingroup$
Thanks @darijgrinberg I will try that out, but since I don't know $T^*$ explicitly will that lead to issues? But I will try this right now and see what I get!
$endgroup$
– InsigMath
Apr 2 at 1:07
$begingroup$
No, just rewrite $T^*(f) = a + bt$ every time you encounter $T^*(f)$.
$endgroup$
– darij grinberg
Apr 2 at 1:07
|
show 1 more comment
$begingroup$
I am taking an linear algebra class and we are on inner product space. I have a question that wants us to evaluate $T^*$ at a given vector. This is not a homework question, it is a textbook question, specifically question 3c in 6.3 of Linear Algebra by Friedberg, Insel, Spence (pg 366). I will it write out though:
$V=P_1(R)$ (the set of polynomials with degree $leq 1$ with real coefficients) with $langle f,grangle=int_-1^1f(t)g(t)dt,$ $T(f)=f'+3f.$ Compute $T^*(f(t))$ where $f(t)=4-2t$.
I know the answer (it's in the textbook) but I am not sure how to obtain the solution. It's my understanding that since $V$ is finite-dimensional we know there is an adjoint and that for all $f,gin P_1(R)$ $langle T(g),frangle=langle g,T^*(f)rangle$. I tried setting $f=4-2t$ and I thought maybe set $g=1$ to make my computation easier but I don't think I am getting this right. I also decided to take the standard basis for $P_1(R)$ and used Gram-Schmidt to obtain an orthonormal basis $gamma = frac1sqrt2,sqrtfrac32x$. I am not really sure what to do next, like do I need to calculate the exact adjoint map or something. Any help would be greatly appreciated.
UPDATE: I figured out based on the comment made by darij grinberg. I used the fact that $langle T(g),frangle=langle g,T^*(f)rangle$ where $g=1$ and $g=t$ and setting $T^*(f) = a+bt$ and figuring out $a$ and $b$ and then using that to compute $T^*(4-2t)$.
linear-algebra linear-transformations
asked Apr 2 at 0:56
InsigMathInsigMath
1,01021021
$endgroup$
I am taking an linear algebra class and we are on inner product space. I have a question that wants us to evaluate $T^*$ at a given vector. This is not a homework question, it is a textbook question, specifically question 3c in 6.3 of Linear Algebra by Friedberg, Insel, Spence (pg 366). I will it write out though:
$V=P_1(R)$ (the set of polynomials with degree $leq 1$ with real coefficients) with $langle f,grangle=int_-1^1f(t)g(t)dt,$ $T(f)=f'+3f.$ Compute $T^*(f(t))$ where $f(t)=4-2t$.
I know the answer (it's in the textbook) but I am not sure how to obtain the solution. It's my understanding that since $V$ is finite-dimensional we know there is an adjoint and that for all $f,gin P_1(R)$ $langle T(g),frangle=langle g,T^*(f)rangle$. I tried setting $f=4-2t$ and I thought maybe set $g=1$ to make my computation easier but I don't think I am getting this right. I also decided to take the standard basis for $P_1(R)$ and used Gram-Schmidt to obtain an orthonormal basis $gamma = frac1sqrt2,sqrtfrac32x$. I am not really sure what to do next, like do I need to calculate the exact adjoint map or something. Any help would be greatly appreciated.
UPDATE: I figured out based on the comment made by darij grinberg. I used the fact that $langle T(g),frangle=langle g,T^*(f)rangle$ where $g=1$ and $g=t$ and setting $T^*(f) = a+bt$ and figuring out $a$ and $b$ and then using that to compute $T^*(4-2t)$.
linear-algebra linear-transformations
linear-algebra linear-transformations
asked Apr 2 at 0:56
InsigMathInsigMath
1,01021021
asked Apr 2 at 0:56
InsigMathInsigMath
1,01021021
edited Apr 2 at 1:26
InsigMath
asked Apr 2 at 0:56
InsigMathInsigMath
1,01021021
asked Apr 2 at 0:56
InsigMathInsigMath
1,01021021
asked Apr 2 at 0:56
InsigMathInsigMath
1,01021021
1,01021021
$begingroup$
What does $P_1left(Rright)$ mean? Polynomials of degree $leq 1$ with real coefficients?
$endgroup$
– darij grinberg
Apr 2 at 1:00
$begingroup$
Oh yes, sorry exactly @darijgrinberg
$endgroup$
– InsigMath
Apr 2 at 1:01
2
$begingroup$
The simplest way to compute $T^*(f)$ is probably to set $T^*(f) = a + bt$ with unknown constants $a$ and $b$, and then solve the system of the two linear equations $left<T(g), fright> = left<g, T^*(f)right>$ for $g = 1$ and $g = t$. (You have to rewrite these equations as explicit linear equations in $a$ and $b$, but you can do this, since the bilinear form is given as an easily computable integral.)
$endgroup$
– darij grinberg
Apr 2 at 1:04
$begingroup$
Thanks @darijgrinberg I will try that out, but since I don't know $T^*$ explicitly will that lead to issues? But I will try this right now and see what I get!
$endgroup$
– InsigMath
Apr 2 at 1:07
$begingroup$
No, just rewrite $T^*(f) = a + bt$ every time you encounter $T^*(f)$.
$endgroup$
– darij grinberg
Apr 2 at 1:07
|
show 1 more comment
$begingroup$
What does $P_1left(Rright)$ mean? Polynomials of degree $leq 1$ with real coefficients?
$endgroup$
– darij grinberg
Apr 2 at 1:00
$begingroup$
Oh yes, sorry exactly @darijgrinberg
$endgroup$
– InsigMath
Apr 2 at 1:01
2
$begingroup$
The simplest way to compute $T^*(f)$ is probably to set $T^*(f) = a + bt$ with unknown constants $a$ and $b$, and then solve the system of the two linear equations $left<T(g), fright> = left<g, T^*(f)right>$ for $g = 1$ and $g = t$. (You have to rewrite these equations as explicit linear equations in $a$ and $b$, but you can do this, since the bilinear form is given as an easily computable integral.)
$endgroup$
– darij grinberg
Apr 2 at 1:04
$begingroup$
Thanks @darijgrinberg I will try that out, but since I don't know $T^*$ explicitly will that lead to issues? But I will try this right now and see what I get!
$endgroup$
– InsigMath
Apr 2 at 1:07
$begingroup$
No, just rewrite $T^*(f) = a + bt$ every time you encounter $T^*(f)$.
$endgroup$
– darij grinberg
Apr 2 at 1:07
$begingroup$
What does $P_1left(Rright)$ mean? Polynomials of degree $leq 1$ with real coefficients?
$endgroup$
– darij grinberg
Apr 2 at 1:00
$begingroup$
What does $P_1left(Rright)$ mean? Polynomials of degree $leq 1$ with real coefficients?
$endgroup$
– darij grinberg
Apr 2 at 1:00
$begingroup$
Oh yes, sorry exactly @darijgrinberg
$endgroup$
– InsigMath
Apr 2 at 1:01
$begingroup$
Oh yes, sorry exactly @darijgrinberg
$endgroup$
– InsigMath
Apr 2 at 1:01
2
2
$begingroup$
The simplest way to compute $T^*(f)$ is probably to set $T^*(f) = a + bt$ with unknown constants $a$ and $b$, and then solve the system of the two linear equations $left<T(g), fright> = left<g, T^*(f)right>$ for $g = 1$ and $g = t$. (You have to rewrite these equations as explicit linear equations in $a$ and $b$, but you can do this, since the bilinear form is given as an easily computable integral.)
$endgroup$
– darij grinberg
Apr 2 at 1:04
$begingroup$
The simplest way to compute $T^*(f)$ is probably to set $T^*(f) = a + bt$ with unknown constants $a$ and $b$, and then solve the system of the two linear equations $left<T(g), fright> = left<g, T^*(f)right>$ for $g = 1$ and $g = t$. (You have to rewrite these equations as explicit linear equations in $a$ and $b$, but you can do this, since the bilinear form is given as an easily computable integral.)
$endgroup$
– darij grinberg
Apr 2 at 1:04
$begingroup$
Thanks @darijgrinberg I will try that out, but since I don't know $T^*$ explicitly will that lead to issues? But I will try this right now and see what I get!
$endgroup$
– InsigMath
Apr 2 at 1:07
$begingroup$
Thanks @darijgrinberg I will try that out, but since I don't know $T^*$ explicitly will that lead to issues? But I will try this right now and see what I get!
$endgroup$
– InsigMath
Apr 2 at 1:07
$begingroup$
No, just rewrite $T^*(f) = a + bt$ every time you encounter $T^*(f)$.
$endgroup$
– darij grinberg
Apr 2 at 1:07
$begingroup$
No, just rewrite $T^*(f) = a + bt$ every time you encounter $T^*(f)$.
$endgroup$
– darij grinberg
Apr 2 at 1:07
|
show 1 more comment
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$begingroup$
What does $P_1left(Rright)$ mean? Polynomials of degree $leq 1$ with real coefficients?
$endgroup$
– darij grinberg
Apr 2 at 1:00
$begingroup$
Oh yes, sorry exactly @darijgrinberg
$endgroup$
– InsigMath
Apr 2 at 1:01
2
$begingroup$
The simplest way to compute $T^*(f)$ is probably to set $T^*(f) = a + bt$ with unknown constants $a$ and $b$, and then solve the system of the two linear equations $left<T(g), fright> = left<g, T^*(f)right>$ for $g = 1$ and $g = t$. (You have to rewrite these equations as explicit linear equations in $a$ and $b$, but you can do this, since the bilinear form is given as an easily computable integral.)
$endgroup$
– darij grinberg
Apr 2 at 1:04
$begingroup$
Thanks @darijgrinberg I will try that out, but since I don't know $T^*$ explicitly will that lead to issues? But I will try this right now and see what I get!
$endgroup$
– InsigMath
Apr 2 at 1:07
$begingroup$
No, just rewrite $T^*(f) = a + bt$ every time you encounter $T^*(f)$.
$endgroup$
– darij grinberg
Apr 2 at 1:07