Finding matrix relative to a polynomial basis and basis in R2?











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Let U be the space of polynomials with basis E=[1,t,t^2], let V=R2 have the basis F=[(1,0),(1,1)] and define a linear map T: U->V by T(f)= (f(3),f'(3)).




  1. Determine the matrix representing T relative to the basis E and F.


  2. What is the Rank of T?


  3. Exhibit a basis for the kernel of T.



I understand how to determine a matrix relative to a single basis in R2, but I've never done it relative to two basis, and never with a polynomial as a basis. Can anyone walk me though how to go about solving this type of problem?










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    – K Split X
    Mar 18 at 23:53















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Let U be the space of polynomials with basis E=[1,t,t^2], let V=R2 have the basis F=[(1,0),(1,1)] and define a linear map T: U->V by T(f)= (f(3),f'(3)).




  1. Determine the matrix representing T relative to the basis E and F.


  2. What is the Rank of T?


  3. Exhibit a basis for the kernel of T.



I understand how to determine a matrix relative to a single basis in R2, but I've never done it relative to two basis, and never with a polynomial as a basis. Can anyone walk me though how to go about solving this type of problem?










share|cite|improve this question






















  • Please use mathjax to format your question
    – K Split X
    Mar 18 at 23:53













up vote
0
down vote

favorite









up vote
0
down vote

favorite











Let U be the space of polynomials with basis E=[1,t,t^2], let V=R2 have the basis F=[(1,0),(1,1)] and define a linear map T: U->V by T(f)= (f(3),f'(3)).




  1. Determine the matrix representing T relative to the basis E and F.


  2. What is the Rank of T?


  3. Exhibit a basis for the kernel of T.



I understand how to determine a matrix relative to a single basis in R2, but I've never done it relative to two basis, and never with a polynomial as a basis. Can anyone walk me though how to go about solving this type of problem?










share|cite|improve this question













Let U be the space of polynomials with basis E=[1,t,t^2], let V=R2 have the basis F=[(1,0),(1,1)] and define a linear map T: U->V by T(f)= (f(3),f'(3)).




  1. Determine the matrix representing T relative to the basis E and F.


  2. What is the Rank of T?


  3. Exhibit a basis for the kernel of T.



I understand how to determine a matrix relative to a single basis in R2, but I've never done it relative to two basis, and never with a polynomial as a basis. Can anyone walk me though how to go about solving this type of problem?







linear-algebra matrices polynomials linear-transformations matrix-rank






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asked Mar 18 at 23:45









Brandon Grothe

81




81












  • Please use mathjax to format your question
    – K Split X
    Mar 18 at 23:53


















  • Please use mathjax to format your question
    – K Split X
    Mar 18 at 23:53
















Please use mathjax to format your question
– K Split X
Mar 18 at 23:53




Please use mathjax to format your question
– K Split X
Mar 18 at 23:53










1 Answer
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HINT



Polynomial of degree 2 in this case are a vector space in the sense that at each polynomial $f(t)=a+bt+ct^2$ we can associate the vector $v=(a,b,c)$.



The linear map $T$ maps a polynomial $f(t)=a+bt+ct^2$ in $(a+3b+9c,b+6c)$. Since we are requeste to use the basis F we need to write $(a+3b+9c,b+6c)$ in this basis that is




  • $x+y=a+3b+9cimplies x=a+3b+9c-b-6c=a+2b+3c$

  • $y= b+6c$


Then



$$T(a,b,c)=(a+2b+3c,b+6c)$$



Can you proceed from here?






share|cite|improve this answer





















  • Thank you, but I'm having trouble wrapping my head around it... could you go a step further?
    – Brandon Grothe
    Mar 19 at 0:11










  • Can you derive the matrix for T from the last step?
    – gimusi
    Mar 19 at 9:12










  • I know I need to turn that into a system of linear equations but I'm confused by the x + y part and how to use it to get to the matrix
    – Brandon Grothe
    Mar 19 at 12:04










  • @BrandonGrothe Did you solve the problem? the matrix representation in the standard basis is $begin{pmatrix}1&3&9\0&1&6end{pmatrix}$ with the bais F is $begin{pmatrix}1&2&3\0&1&6end{pmatrix}$
    – gimusi
    Mar 23 at 13:32













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1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
0
down vote



accepted










HINT



Polynomial of degree 2 in this case are a vector space in the sense that at each polynomial $f(t)=a+bt+ct^2$ we can associate the vector $v=(a,b,c)$.



The linear map $T$ maps a polynomial $f(t)=a+bt+ct^2$ in $(a+3b+9c,b+6c)$. Since we are requeste to use the basis F we need to write $(a+3b+9c,b+6c)$ in this basis that is




  • $x+y=a+3b+9cimplies x=a+3b+9c-b-6c=a+2b+3c$

  • $y= b+6c$


Then



$$T(a,b,c)=(a+2b+3c,b+6c)$$



Can you proceed from here?






share|cite|improve this answer





















  • Thank you, but I'm having trouble wrapping my head around it... could you go a step further?
    – Brandon Grothe
    Mar 19 at 0:11










  • Can you derive the matrix for T from the last step?
    – gimusi
    Mar 19 at 9:12










  • I know I need to turn that into a system of linear equations but I'm confused by the x + y part and how to use it to get to the matrix
    – Brandon Grothe
    Mar 19 at 12:04










  • @BrandonGrothe Did you solve the problem? the matrix representation in the standard basis is $begin{pmatrix}1&3&9\0&1&6end{pmatrix}$ with the bais F is $begin{pmatrix}1&2&3\0&1&6end{pmatrix}$
    – gimusi
    Mar 23 at 13:32

















up vote
0
down vote



accepted










HINT



Polynomial of degree 2 in this case are a vector space in the sense that at each polynomial $f(t)=a+bt+ct^2$ we can associate the vector $v=(a,b,c)$.



The linear map $T$ maps a polynomial $f(t)=a+bt+ct^2$ in $(a+3b+9c,b+6c)$. Since we are requeste to use the basis F we need to write $(a+3b+9c,b+6c)$ in this basis that is




  • $x+y=a+3b+9cimplies x=a+3b+9c-b-6c=a+2b+3c$

  • $y= b+6c$


Then



$$T(a,b,c)=(a+2b+3c,b+6c)$$



Can you proceed from here?






share|cite|improve this answer





















  • Thank you, but I'm having trouble wrapping my head around it... could you go a step further?
    – Brandon Grothe
    Mar 19 at 0:11










  • Can you derive the matrix for T from the last step?
    – gimusi
    Mar 19 at 9:12










  • I know I need to turn that into a system of linear equations but I'm confused by the x + y part and how to use it to get to the matrix
    – Brandon Grothe
    Mar 19 at 12:04










  • @BrandonGrothe Did you solve the problem? the matrix representation in the standard basis is $begin{pmatrix}1&3&9\0&1&6end{pmatrix}$ with the bais F is $begin{pmatrix}1&2&3\0&1&6end{pmatrix}$
    – gimusi
    Mar 23 at 13:32















up vote
0
down vote



accepted







up vote
0
down vote



accepted






HINT



Polynomial of degree 2 in this case are a vector space in the sense that at each polynomial $f(t)=a+bt+ct^2$ we can associate the vector $v=(a,b,c)$.



The linear map $T$ maps a polynomial $f(t)=a+bt+ct^2$ in $(a+3b+9c,b+6c)$. Since we are requeste to use the basis F we need to write $(a+3b+9c,b+6c)$ in this basis that is




  • $x+y=a+3b+9cimplies x=a+3b+9c-b-6c=a+2b+3c$

  • $y= b+6c$


Then



$$T(a,b,c)=(a+2b+3c,b+6c)$$



Can you proceed from here?






share|cite|improve this answer












HINT



Polynomial of degree 2 in this case are a vector space in the sense that at each polynomial $f(t)=a+bt+ct^2$ we can associate the vector $v=(a,b,c)$.



The linear map $T$ maps a polynomial $f(t)=a+bt+ct^2$ in $(a+3b+9c,b+6c)$. Since we are requeste to use the basis F we need to write $(a+3b+9c,b+6c)$ in this basis that is




  • $x+y=a+3b+9cimplies x=a+3b+9c-b-6c=a+2b+3c$

  • $y= b+6c$


Then



$$T(a,b,c)=(a+2b+3c,b+6c)$$



Can you proceed from here?







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Mar 18 at 23:59









gimusi

89.7k74495




89.7k74495












  • Thank you, but I'm having trouble wrapping my head around it... could you go a step further?
    – Brandon Grothe
    Mar 19 at 0:11










  • Can you derive the matrix for T from the last step?
    – gimusi
    Mar 19 at 9:12










  • I know I need to turn that into a system of linear equations but I'm confused by the x + y part and how to use it to get to the matrix
    – Brandon Grothe
    Mar 19 at 12:04










  • @BrandonGrothe Did you solve the problem? the matrix representation in the standard basis is $begin{pmatrix}1&3&9\0&1&6end{pmatrix}$ with the bais F is $begin{pmatrix}1&2&3\0&1&6end{pmatrix}$
    – gimusi
    Mar 23 at 13:32




















  • Thank you, but I'm having trouble wrapping my head around it... could you go a step further?
    – Brandon Grothe
    Mar 19 at 0:11










  • Can you derive the matrix for T from the last step?
    – gimusi
    Mar 19 at 9:12










  • I know I need to turn that into a system of linear equations but I'm confused by the x + y part and how to use it to get to the matrix
    – Brandon Grothe
    Mar 19 at 12:04










  • @BrandonGrothe Did you solve the problem? the matrix representation in the standard basis is $begin{pmatrix}1&3&9\0&1&6end{pmatrix}$ with the bais F is $begin{pmatrix}1&2&3\0&1&6end{pmatrix}$
    – gimusi
    Mar 23 at 13:32


















Thank you, but I'm having trouble wrapping my head around it... could you go a step further?
– Brandon Grothe
Mar 19 at 0:11




Thank you, but I'm having trouble wrapping my head around it... could you go a step further?
– Brandon Grothe
Mar 19 at 0:11












Can you derive the matrix for T from the last step?
– gimusi
Mar 19 at 9:12




Can you derive the matrix for T from the last step?
– gimusi
Mar 19 at 9:12












I know I need to turn that into a system of linear equations but I'm confused by the x + y part and how to use it to get to the matrix
– Brandon Grothe
Mar 19 at 12:04




I know I need to turn that into a system of linear equations but I'm confused by the x + y part and how to use it to get to the matrix
– Brandon Grothe
Mar 19 at 12:04












@BrandonGrothe Did you solve the problem? the matrix representation in the standard basis is $begin{pmatrix}1&3&9\0&1&6end{pmatrix}$ with the bais F is $begin{pmatrix}1&2&3\0&1&6end{pmatrix}$
– gimusi
Mar 23 at 13:32






@BrandonGrothe Did you solve the problem? the matrix representation in the standard basis is $begin{pmatrix}1&3&9\0&1&6end{pmatrix}$ with the bais F is $begin{pmatrix}1&2&3\0&1&6end{pmatrix}$
– gimusi
Mar 23 at 13:32




















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