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write coordinates in the base of $1,x,x^2,x^3$



Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Decide a subspaceRemind me: How do we convert coordinates to space with different basis and origin?Linear transformation with matrices in baseRepresentation of Functionals of a vector spaceCalculating vector coordinates with new baseIf $B= ((X-1)^2, X-1, 1)$ is a base of $mathbbR_2[X]$, how do I the coordinates of a polynomial in this base?A base for the kernel of a definite integral?Express the class of the polynomial $3x^2 + 2x - 2$Find vector coordinates with this information?Linear transformation via taking derivatives, what are the matrices?










0












$begingroup$


I am given this task and have no clue what to do.



We are given, linear space,which is spanned by polynomials (with real coefficients), whose degrees are at most 3, and we are give the polynomial $f(x) =2-3x+x^3$



How can I write $f(x)$ coordinates in the base $1,x,x^2,x^3$?










share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    Can you write $2-3x+x^3$ as a linear combination of $1, x, x^2, x^3$?
    $endgroup$
    – J. W. Tanner
    Mar 27 at 17:51










  • $begingroup$
    Welcome to the website. Refer here for a quick guide on Mathjax to typeset your Math equations in the future.
    $endgroup$
    – Shubham Johri
    Mar 27 at 17:53















0












$begingroup$


I am given this task and have no clue what to do.



We are given, linear space,which is spanned by polynomials (with real coefficients), whose degrees are at most 3, and we are give the polynomial $f(x) =2-3x+x^3$



How can I write $f(x)$ coordinates in the base $1,x,x^2,x^3$?










share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    Can you write $2-3x+x^3$ as a linear combination of $1, x, x^2, x^3$?
    $endgroup$
    – J. W. Tanner
    Mar 27 at 17:51










  • $begingroup$
    Welcome to the website. Refer here for a quick guide on Mathjax to typeset your Math equations in the future.
    $endgroup$
    – Shubham Johri
    Mar 27 at 17:53













0












0








0


1



$begingroup$


I am given this task and have no clue what to do.



We are given, linear space,which is spanned by polynomials (with real coefficients), whose degrees are at most 3, and we are give the polynomial $f(x) =2-3x+x^3$



How can I write $f(x)$ coordinates in the base $1,x,x^2,x^3$?










share|cite|improve this question











$endgroup$




I am given this task and have no clue what to do.



We are given, linear space,which is spanned by polynomials (with real coefficients), whose degrees are at most 3, and we are give the polynomial $f(x) =2-3x+x^3$



How can I write $f(x)$ coordinates in the base $1,x,x^2,x^3$?







linear-algebra






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 27 at 17:48









Shubham Johri

5,668918




5,668918










asked Mar 27 at 17:45









EdvardsZEdvardsZ

31




31







  • 1




    $begingroup$
    Can you write $2-3x+x^3$ as a linear combination of $1, x, x^2, x^3$?
    $endgroup$
    – J. W. Tanner
    Mar 27 at 17:51










  • $begingroup$
    Welcome to the website. Refer here for a quick guide on Mathjax to typeset your Math equations in the future.
    $endgroup$
    – Shubham Johri
    Mar 27 at 17:53












  • 1




    $begingroup$
    Can you write $2-3x+x^3$ as a linear combination of $1, x, x^2, x^3$?
    $endgroup$
    – J. W. Tanner
    Mar 27 at 17:51










  • $begingroup$
    Welcome to the website. Refer here for a quick guide on Mathjax to typeset your Math equations in the future.
    $endgroup$
    – Shubham Johri
    Mar 27 at 17:53







1




1




$begingroup$
Can you write $2-3x+x^3$ as a linear combination of $1, x, x^2, x^3$?
$endgroup$
– J. W. Tanner
Mar 27 at 17:51




$begingroup$
Can you write $2-3x+x^3$ as a linear combination of $1, x, x^2, x^3$?
$endgroup$
– J. W. Tanner
Mar 27 at 17:51












$begingroup$
Welcome to the website. Refer here for a quick guide on Mathjax to typeset your Math equations in the future.
$endgroup$
– Shubham Johri
Mar 27 at 17:53




$begingroup$
Welcome to the website. Refer here for a quick guide on Mathjax to typeset your Math equations in the future.
$endgroup$
– Shubham Johri
Mar 27 at 17:53










2 Answers
2






active

oldest

votes


















0












$begingroup$

When in doubt, go back to basic definitions. What does it mean for the tuple $(a,b,c,d)$ to be the coordinates of a vector $mathbf v$ relative to some ordered basis $mathcal B=(mathbf v_1, mathbf v_2,mathbf v_3,mathbf v_4)$? It means that $mathbf v$ can be expressed uniquely as the linear combination $amathbf v_1+bmathbf v_2+cmathbf v_3+dmathbf v_4$, i.e., the coordinates of $mathbf v$ relative to $mathcal B$ are exactly the coefficients of the basis vectors in that unique linear combination.



Knowing this, what do you think the coordinates of the polynomial $a+bx+cx^2+dx^3$ relative to the ordered basis $(1,x,x^2,x^3)$ are?






share|cite|improve this answer









$endgroup$




















    0












    $begingroup$

    Hint



    Let the co-ordinates of $f(x)$ with respect to the (ordered) basis $1,x,x^2,x^3$ be $(a,b,c,d)$. This means $f(x)=2-3x+x^3=a+bx+cx^2+dx^3$. Now compare the coefficients of different powers of $x$ on each side.






    share|cite|improve this answer









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      2 Answers
      2






      active

      oldest

      votes








      2 Answers
      2






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      0












      $begingroup$

      When in doubt, go back to basic definitions. What does it mean for the tuple $(a,b,c,d)$ to be the coordinates of a vector $mathbf v$ relative to some ordered basis $mathcal B=(mathbf v_1, mathbf v_2,mathbf v_3,mathbf v_4)$? It means that $mathbf v$ can be expressed uniquely as the linear combination $amathbf v_1+bmathbf v_2+cmathbf v_3+dmathbf v_4$, i.e., the coordinates of $mathbf v$ relative to $mathcal B$ are exactly the coefficients of the basis vectors in that unique linear combination.



      Knowing this, what do you think the coordinates of the polynomial $a+bx+cx^2+dx^3$ relative to the ordered basis $(1,x,x^2,x^3)$ are?






      share|cite|improve this answer









      $endgroup$

















        0












        $begingroup$

        When in doubt, go back to basic definitions. What does it mean for the tuple $(a,b,c,d)$ to be the coordinates of a vector $mathbf v$ relative to some ordered basis $mathcal B=(mathbf v_1, mathbf v_2,mathbf v_3,mathbf v_4)$? It means that $mathbf v$ can be expressed uniquely as the linear combination $amathbf v_1+bmathbf v_2+cmathbf v_3+dmathbf v_4$, i.e., the coordinates of $mathbf v$ relative to $mathcal B$ are exactly the coefficients of the basis vectors in that unique linear combination.



        Knowing this, what do you think the coordinates of the polynomial $a+bx+cx^2+dx^3$ relative to the ordered basis $(1,x,x^2,x^3)$ are?






        share|cite|improve this answer









        $endgroup$















          0












          0








          0





          $begingroup$

          When in doubt, go back to basic definitions. What does it mean for the tuple $(a,b,c,d)$ to be the coordinates of a vector $mathbf v$ relative to some ordered basis $mathcal B=(mathbf v_1, mathbf v_2,mathbf v_3,mathbf v_4)$? It means that $mathbf v$ can be expressed uniquely as the linear combination $amathbf v_1+bmathbf v_2+cmathbf v_3+dmathbf v_4$, i.e., the coordinates of $mathbf v$ relative to $mathcal B$ are exactly the coefficients of the basis vectors in that unique linear combination.



          Knowing this, what do you think the coordinates of the polynomial $a+bx+cx^2+dx^3$ relative to the ordered basis $(1,x,x^2,x^3)$ are?






          share|cite|improve this answer









          $endgroup$



          When in doubt, go back to basic definitions. What does it mean for the tuple $(a,b,c,d)$ to be the coordinates of a vector $mathbf v$ relative to some ordered basis $mathcal B=(mathbf v_1, mathbf v_2,mathbf v_3,mathbf v_4)$? It means that $mathbf v$ can be expressed uniquely as the linear combination $amathbf v_1+bmathbf v_2+cmathbf v_3+dmathbf v_4$, i.e., the coordinates of $mathbf v$ relative to $mathcal B$ are exactly the coefficients of the basis vectors in that unique linear combination.



          Knowing this, what do you think the coordinates of the polynomial $a+bx+cx^2+dx^3$ relative to the ordered basis $(1,x,x^2,x^3)$ are?







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Mar 27 at 18:29









          amdamd

          31.9k21053




          31.9k21053





















              0












              $begingroup$

              Hint



              Let the co-ordinates of $f(x)$ with respect to the (ordered) basis $1,x,x^2,x^3$ be $(a,b,c,d)$. This means $f(x)=2-3x+x^3=a+bx+cx^2+dx^3$. Now compare the coefficients of different powers of $x$ on each side.






              share|cite|improve this answer









              $endgroup$

















                0












                $begingroup$

                Hint



                Let the co-ordinates of $f(x)$ with respect to the (ordered) basis $1,x,x^2,x^3$ be $(a,b,c,d)$. This means $f(x)=2-3x+x^3=a+bx+cx^2+dx^3$. Now compare the coefficients of different powers of $x$ on each side.






                share|cite|improve this answer









                $endgroup$















                  0












                  0








                  0





                  $begingroup$

                  Hint



                  Let the co-ordinates of $f(x)$ with respect to the (ordered) basis $1,x,x^2,x^3$ be $(a,b,c,d)$. This means $f(x)=2-3x+x^3=a+bx+cx^2+dx^3$. Now compare the coefficients of different powers of $x$ on each side.






                  share|cite|improve this answer









                  $endgroup$



                  Hint



                  Let the co-ordinates of $f(x)$ with respect to the (ordered) basis $1,x,x^2,x^3$ be $(a,b,c,d)$. This means $f(x)=2-3x+x^3=a+bx+cx^2+dx^3$. Now compare the coefficients of different powers of $x$ on each side.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Mar 27 at 17:51









                  Shubham JohriShubham Johri

                  5,668918




                  5,668918



























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