Find the matrix of $T$ with respect to standard basis of $V$.












0












$begingroup$


Let $T:Vto V$ be the rotation by an angle $theta$ counterclockwise in the plane passing through the origin perpendicular to $(1,2,3)$ where $V=Bbb R^3$



Find the matrix of $T$ with respect to standard basis of $V$.



I know that the equation of the plane passing through the origin perpendicular to $(1,2,3)$ is $x+2y+3z=0$ but I dont know how to find the matrix .



I really dont understand where does $T$ map the vector $(1,0,0)$ .



Can someone kindly help me?



Hints will suffice.I dont see any way out






linear-algebra geometry linear-transformations







share|cite|improve this question









$endgroup$

















    0












    $begingroup$


    Let $T:Vto V$ be the rotation by an angle $theta$ counterclockwise in the plane passing through the origin perpendicular to $(1,2,3)$ where $V=Bbb R^3$



    Find the matrix of $T$ with respect to standard basis of $V$.



    I know that the equation of the plane passing through the origin perpendicular to $(1,2,3)$ is $x+2y+3z=0$ but I dont know how to find the matrix .



    I really dont understand where does $T$ map the vector $(1,0,0)$ .



    Can someone kindly help me?



    Hints will suffice.I dont see any way out






    linear-algebra geometry linear-transformations







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Let $T:Vto V$ be the rotation by an angle $theta$ counterclockwise in the plane passing through the origin perpendicular to $(1,2,3)$ where $V=Bbb R^3$



      Find the matrix of $T$ with respect to standard basis of $V$.



      I know that the equation of the plane passing through the origin perpendicular to $(1,2,3)$ is $x+2y+3z=0$ but I dont know how to find the matrix .



      I really dont understand where does $T$ map the vector $(1,0,0)$ .



      Can someone kindly help me?



      Hints will suffice.I dont see any way out






      linear-algebra geometry linear-transformations







      share|cite|improve this question









      $endgroup$




      Let $T:Vto V$ be the rotation by an angle $theta$ counterclockwise in the plane passing through the origin perpendicular to $(1,2,3)$ where $V=Bbb R^3$



      Find the matrix of $T$ with respect to standard basis of $V$.



      I know that the equation of the plane passing through the origin perpendicular to $(1,2,3)$ is $x+2y+3z=0$ but I dont know how to find the matrix .



      I really dont understand where does $T$ map the vector $(1,0,0)$ .



      Can someone kindly help me?



      Hints will suffice.I dont see any way out






      linear-algebra geometry linear-transformations




      linear-algebra geometry linear-transformations






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 6 '18 at 9:38









      Join_PhDJoin_PhD

      4018




      4018






















          1 Answer
          1






          active

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          1












          $begingroup$

          Extend $v_1=frac{1}{sqrt{14}} (1,2,3)$ to an orthonormal basis $B=(v_1, v_2, v_3)$. Write down the matrix of $T$ with respect to this basis. Perform a change of basis to the standard basis $S=(e_1, e_2, e_3)$.




          The vector $v_1$ I picked is a vector in the direction $(1,2,3)$ having length $|v_1|=1$. To extend this to an orthonormal basis, you can proceed in (at least) two ways:




          1. extend to any basis first, e.g. $v_1, e_1, e_2$ and apply Gram–Schmidt orhonormalization

          2. find one unit vector $v_2$ orthogonal to $v_1$ and then compute the cross product $v_3=v_1times v_2$.



          Having done that, we can find the matrix of $T$ with respect to $B=(v_1, v_2, v_3)$.



          Since $v_1$ is a vector along the axis of rotation, we have $T(v_1)=v_1$. The vectors $v_2,v_3$ form an orthonormal basis of the plane perpendicular to $v_1$, so they get rotated by $theta$ in this plane and we have
          begin{align*}
          T(v_2) &= cos(theta), v_2 + sin(theta), v_3, \
          T(v_2) &= -sin(theta), v_2 + cos(theta), v_3.
          end{align*}
          Thus, the matrix of $T$ with respect to the basis $B$ is given by
          $$M(T)_B = begin{bmatrix} 1 & 0&0\0 &cos theta&-sin theta\0& sintheta &costhetaend{bmatrix}.$$
          The change of base matrix from $v_1,v_2,v_3$ to the standard basis is the matrix $C_{S,B}=(v_1 ,|, v_2 ,|, v_3)$ with columns $v_1, v_2, v_3$.



          Hence, the matrix of $T$ with respect to the standard basis is
          $$
          M(T)_S = C_{S,B} ,M(T)_B ,C_{B,S} = C_{S,B} ,M(T)_B ,C_{S,B}^{-1}.
          $$






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            But T is the rotation about this plane,I dont understand why should I follow the above procedure to find its matrix
            $endgroup$
            – Join_PhD
            Dec 6 '18 at 10:08










          • $begingroup$
            Can you please give some details using geometry
            $endgroup$
            – Join_PhD
            Dec 6 '18 at 10:08










          • $begingroup$
            Do you know how to write down a rotation about the axis $(1,0,0)$ by an angle $theta$?
            $endgroup$
            – Christoph
            Dec 6 '18 at 10:09










          • $begingroup$
            Yes it is $begin{bmatrix} 1 & 0&0\0 &cos theta&-sin theta\0& sintheta &costhetaend{bmatrix}$
            $endgroup$
            – Join_PhD
            Dec 6 '18 at 10:10












          • $begingroup$
            This is the matrix of $T$ with respect to $v_1, v_2, v_3$. Now perform a change of basis.
            $endgroup$
            – Christoph
            Dec 6 '18 at 10:12













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          1












          $begingroup$

          Extend $v_1=frac{1}{sqrt{14}} (1,2,3)$ to an orthonormal basis $B=(v_1, v_2, v_3)$. Write down the matrix of $T$ with respect to this basis. Perform a change of basis to the standard basis $S=(e_1, e_2, e_3)$.




          The vector $v_1$ I picked is a vector in the direction $(1,2,3)$ having length $|v_1|=1$. To extend this to an orthonormal basis, you can proceed in (at least) two ways:




          1. extend to any basis first, e.g. $v_1, e_1, e_2$ and apply Gram–Schmidt orhonormalization

          2. find one unit vector $v_2$ orthogonal to $v_1$ and then compute the cross product $v_3=v_1times v_2$.



          Having done that, we can find the matrix of $T$ with respect to $B=(v_1, v_2, v_3)$.



          Since $v_1$ is a vector along the axis of rotation, we have $T(v_1)=v_1$. The vectors $v_2,v_3$ form an orthonormal basis of the plane perpendicular to $v_1$, so they get rotated by $theta$ in this plane and we have
          begin{align*}
          T(v_2) &= cos(theta), v_2 + sin(theta), v_3, \
          T(v_2) &= -sin(theta), v_2 + cos(theta), v_3.
          end{align*}
          Thus, the matrix of $T$ with respect to the basis $B$ is given by
          $$M(T)_B = begin{bmatrix} 1 & 0&0\0 &cos theta&-sin theta\0& sintheta &costhetaend{bmatrix}.$$
          The change of base matrix from $v_1,v_2,v_3$ to the standard basis is the matrix $C_{S,B}=(v_1 ,|, v_2 ,|, v_3)$ with columns $v_1, v_2, v_3$.



          Hence, the matrix of $T$ with respect to the standard basis is
          $$
          M(T)_S = C_{S,B} ,M(T)_B ,C_{B,S} = C_{S,B} ,M(T)_B ,C_{S,B}^{-1}.
          $$






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            But T is the rotation about this plane,I dont understand why should I follow the above procedure to find its matrix
            $endgroup$
            – Join_PhD
            Dec 6 '18 at 10:08










          • $begingroup$
            Can you please give some details using geometry
            $endgroup$
            – Join_PhD
            Dec 6 '18 at 10:08










          • $begingroup$
            Do you know how to write down a rotation about the axis $(1,0,0)$ by an angle $theta$?
            $endgroup$
            – Christoph
            Dec 6 '18 at 10:09










          • $begingroup$
            Yes it is $begin{bmatrix} 1 & 0&0\0 &cos theta&-sin theta\0& sintheta &costhetaend{bmatrix}$
            $endgroup$
            – Join_PhD
            Dec 6 '18 at 10:10












          • $begingroup$
            This is the matrix of $T$ with respect to $v_1, v_2, v_3$. Now perform a change of basis.
            $endgroup$
            – Christoph
            Dec 6 '18 at 10:12


















          1












          $begingroup$

          Extend $v_1=frac{1}{sqrt{14}} (1,2,3)$ to an orthonormal basis $B=(v_1, v_2, v_3)$. Write down the matrix of $T$ with respect to this basis. Perform a change of basis to the standard basis $S=(e_1, e_2, e_3)$.




          The vector $v_1$ I picked is a vector in the direction $(1,2,3)$ having length $|v_1|=1$. To extend this to an orthonormal basis, you can proceed in (at least) two ways:




          1. extend to any basis first, e.g. $v_1, e_1, e_2$ and apply Gram–Schmidt orhonormalization

          2. find one unit vector $v_2$ orthogonal to $v_1$ and then compute the cross product $v_3=v_1times v_2$.



          Having done that, we can find the matrix of $T$ with respect to $B=(v_1, v_2, v_3)$.



          Since $v_1$ is a vector along the axis of rotation, we have $T(v_1)=v_1$. The vectors $v_2,v_3$ form an orthonormal basis of the plane perpendicular to $v_1$, so they get rotated by $theta$ in this plane and we have
          begin{align*}
          T(v_2) &= cos(theta), v_2 + sin(theta), v_3, \
          T(v_2) &= -sin(theta), v_2 + cos(theta), v_3.
          end{align*}
          Thus, the matrix of $T$ with respect to the basis $B$ is given by
          $$M(T)_B = begin{bmatrix} 1 & 0&0\0 &cos theta&-sin theta\0& sintheta &costhetaend{bmatrix}.$$
          The change of base matrix from $v_1,v_2,v_3$ to the standard basis is the matrix $C_{S,B}=(v_1 ,|, v_2 ,|, v_3)$ with columns $v_1, v_2, v_3$.



          Hence, the matrix of $T$ with respect to the standard basis is
          $$
          M(T)_S = C_{S,B} ,M(T)_B ,C_{B,S} = C_{S,B} ,M(T)_B ,C_{S,B}^{-1}.
          $$






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            But T is the rotation about this plane,I dont understand why should I follow the above procedure to find its matrix
            $endgroup$
            – Join_PhD
            Dec 6 '18 at 10:08










          • $begingroup$
            Can you please give some details using geometry
            $endgroup$
            – Join_PhD
            Dec 6 '18 at 10:08










          • $begingroup$
            Do you know how to write down a rotation about the axis $(1,0,0)$ by an angle $theta$?
            $endgroup$
            – Christoph
            Dec 6 '18 at 10:09










          • $begingroup$
            Yes it is $begin{bmatrix} 1 & 0&0\0 &cos theta&-sin theta\0& sintheta &costhetaend{bmatrix}$
            $endgroup$
            – Join_PhD
            Dec 6 '18 at 10:10












          • $begingroup$
            This is the matrix of $T$ with respect to $v_1, v_2, v_3$. Now perform a change of basis.
            $endgroup$
            – Christoph
            Dec 6 '18 at 10:12
















          1












          1








          1





          $begingroup$

          Extend $v_1=frac{1}{sqrt{14}} (1,2,3)$ to an orthonormal basis $B=(v_1, v_2, v_3)$. Write down the matrix of $T$ with respect to this basis. Perform a change of basis to the standard basis $S=(e_1, e_2, e_3)$.




          The vector $v_1$ I picked is a vector in the direction $(1,2,3)$ having length $|v_1|=1$. To extend this to an orthonormal basis, you can proceed in (at least) two ways:




          1. extend to any basis first, e.g. $v_1, e_1, e_2$ and apply Gram–Schmidt orhonormalization

          2. find one unit vector $v_2$ orthogonal to $v_1$ and then compute the cross product $v_3=v_1times v_2$.



          Having done that, we can find the matrix of $T$ with respect to $B=(v_1, v_2, v_3)$.



          Since $v_1$ is a vector along the axis of rotation, we have $T(v_1)=v_1$. The vectors $v_2,v_3$ form an orthonormal basis of the plane perpendicular to $v_1$, so they get rotated by $theta$ in this plane and we have
          begin{align*}
          T(v_2) &= cos(theta), v_2 + sin(theta), v_3, \
          T(v_2) &= -sin(theta), v_2 + cos(theta), v_3.
          end{align*}
          Thus, the matrix of $T$ with respect to the basis $B$ is given by
          $$M(T)_B = begin{bmatrix} 1 & 0&0\0 &cos theta&-sin theta\0& sintheta &costhetaend{bmatrix}.$$
          The change of base matrix from $v_1,v_2,v_3$ to the standard basis is the matrix $C_{S,B}=(v_1 ,|, v_2 ,|, v_3)$ with columns $v_1, v_2, v_3$.



          Hence, the matrix of $T$ with respect to the standard basis is
          $$
          M(T)_S = C_{S,B} ,M(T)_B ,C_{B,S} = C_{S,B} ,M(T)_B ,C_{S,B}^{-1}.
          $$






          share|cite|improve this answer











          $endgroup$



          Extend $v_1=frac{1}{sqrt{14}} (1,2,3)$ to an orthonormal basis $B=(v_1, v_2, v_3)$. Write down the matrix of $T$ with respect to this basis. Perform a change of basis to the standard basis $S=(e_1, e_2, e_3)$.




          The vector $v_1$ I picked is a vector in the direction $(1,2,3)$ having length $|v_1|=1$. To extend this to an orthonormal basis, you can proceed in (at least) two ways:




          1. extend to any basis first, e.g. $v_1, e_1, e_2$ and apply Gram–Schmidt orhonormalization

          2. find one unit vector $v_2$ orthogonal to $v_1$ and then compute the cross product $v_3=v_1times v_2$.



          Having done that, we can find the matrix of $T$ with respect to $B=(v_1, v_2, v_3)$.



          Since $v_1$ is a vector along the axis of rotation, we have $T(v_1)=v_1$. The vectors $v_2,v_3$ form an orthonormal basis of the plane perpendicular to $v_1$, so they get rotated by $theta$ in this plane and we have
          begin{align*}
          T(v_2) &= cos(theta), v_2 + sin(theta), v_3, \
          T(v_2) &= -sin(theta), v_2 + cos(theta), v_3.
          end{align*}
          Thus, the matrix of $T$ with respect to the basis $B$ is given by
          $$M(T)_B = begin{bmatrix} 1 & 0&0\0 &cos theta&-sin theta\0& sintheta &costhetaend{bmatrix}.$$
          The change of base matrix from $v_1,v_2,v_3$ to the standard basis is the matrix $C_{S,B}=(v_1 ,|, v_2 ,|, v_3)$ with columns $v_1, v_2, v_3$.



          Hence, the matrix of $T$ with respect to the standard basis is
          $$
          M(T)_S = C_{S,B} ,M(T)_B ,C_{B,S} = C_{S,B} ,M(T)_B ,C_{S,B}^{-1}.
          $$







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Dec 6 '18 at 10:35

























          answered Dec 6 '18 at 9:46









          ChristophChristoph

          12k1642




          12k1642












          • $begingroup$
            But T is the rotation about this plane,I dont understand why should I follow the above procedure to find its matrix
            $endgroup$
            – Join_PhD
            Dec 6 '18 at 10:08










          • $begingroup$
            Can you please give some details using geometry
            $endgroup$
            – Join_PhD
            Dec 6 '18 at 10:08










          • $begingroup$
            Do you know how to write down a rotation about the axis $(1,0,0)$ by an angle $theta$?
            $endgroup$
            – Christoph
            Dec 6 '18 at 10:09










          • $begingroup$
            Yes it is $begin{bmatrix} 1 & 0&0\0 &cos theta&-sin theta\0& sintheta &costhetaend{bmatrix}$
            $endgroup$
            – Join_PhD
            Dec 6 '18 at 10:10












          • $begingroup$
            This is the matrix of $T$ with respect to $v_1, v_2, v_3$. Now perform a change of basis.
            $endgroup$
            – Christoph
            Dec 6 '18 at 10:12




















          • $begingroup$
            But T is the rotation about this plane,I dont understand why should I follow the above procedure to find its matrix
            $endgroup$
            – Join_PhD
            Dec 6 '18 at 10:08










          • $begingroup$
            Can you please give some details using geometry
            $endgroup$
            – Join_PhD
            Dec 6 '18 at 10:08










          • $begingroup$
            Do you know how to write down a rotation about the axis $(1,0,0)$ by an angle $theta$?
            $endgroup$
            – Christoph
            Dec 6 '18 at 10:09










          • $begingroup$
            Yes it is $begin{bmatrix} 1 & 0&0\0 &cos theta&-sin theta\0& sintheta &costhetaend{bmatrix}$
            $endgroup$
            – Join_PhD
            Dec 6 '18 at 10:10












          • $begingroup$
            This is the matrix of $T$ with respect to $v_1, v_2, v_3$. Now perform a change of basis.
            $endgroup$
            – Christoph
            Dec 6 '18 at 10:12


















          $begingroup$
          But T is the rotation about this plane,I dont understand why should I follow the above procedure to find its matrix
          $endgroup$
          – Join_PhD
          Dec 6 '18 at 10:08




          $begingroup$
          But T is the rotation about this plane,I dont understand why should I follow the above procedure to find its matrix
          $endgroup$
          – Join_PhD
          Dec 6 '18 at 10:08












          $begingroup$
          Can you please give some details using geometry
          $endgroup$
          – Join_PhD
          Dec 6 '18 at 10:08




          $begingroup$
          Can you please give some details using geometry
          $endgroup$
          – Join_PhD
          Dec 6 '18 at 10:08












          $begingroup$
          Do you know how to write down a rotation about the axis $(1,0,0)$ by an angle $theta$?
          $endgroup$
          – Christoph
          Dec 6 '18 at 10:09




          $begingroup$
          Do you know how to write down a rotation about the axis $(1,0,0)$ by an angle $theta$?
          $endgroup$
          – Christoph
          Dec 6 '18 at 10:09












          $begingroup$
          Yes it is $begin{bmatrix} 1 & 0&0\0 &cos theta&-sin theta\0& sintheta &costhetaend{bmatrix}$
          $endgroup$
          – Join_PhD
          Dec 6 '18 at 10:10






          $begingroup$
          Yes it is $begin{bmatrix} 1 & 0&0\0 &cos theta&-sin theta\0& sintheta &costhetaend{bmatrix}$
          $endgroup$
          – Join_PhD
          Dec 6 '18 at 10:10














          $begingroup$
          This is the matrix of $T$ with respect to $v_1, v_2, v_3$. Now perform a change of basis.
          $endgroup$
          – Christoph
          Dec 6 '18 at 10:12






          $begingroup$
          This is the matrix of $T$ with respect to $v_1, v_2, v_3$. Now perform a change of basis.
          $endgroup$
          – Christoph
          Dec 6 '18 at 10:12




















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          Musa

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          Para el género de plantas, véase Musa (planta). Apolo y la Nueve Musas En la mitología griega, las musas (en griego antiguo μοῦσαι « mousai» ) son, según los escritores más antiguos, las divinidades inspiradoras de las Artes: cada una de ellas está relacionada con ramas artísticas y del conocimiento. Son hijas de Zeus y de Mnemósine, compañeras del séquito de Apolo, dios olímpico de la música y patrón de las bellas artes, quien tuvo romances con cada una de ellas, dejando descendientes. Bajaban a la tierra a susurrar ideas e inspirar a aquellos mortales que las invocaran. En la época más arcaica eran las ninfas inspiradoras de las fuentes, en las cuales eran adoradas. Finalmente, alrededor de los siglos VIII-VII a. C. [ 1 ] ​ prevaleció en todo el territorio de la Hélade la adoración de las nueve Musas, que son Calíope, Clío, Erato, Euterpe, Melpómene, Polimnia, Talía, Terpsícore y Urania. El culto a las musas era originalmente de Tracia y Beocia, y fueron de vital impo...
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