Linear Algebra with Sage

<Matrix Representation>


Made by SKKU Linear Algebra Lab (2011)




Declare a variables and define a linear transformation. (변수 선언과 선형변환 정의)

var('x,y,z,w')

T(x,y,z,w)=(x+2*y, x-3*z+w, 2*y+3*z+4*w);T

(x, y, z, w) |--> (x + 2*y, w+ x - 3*z, 4*w + 2*y + 3*z)


@@\{\textbf{e}_{1}@@, @@\textbf{e}_{2}@@, @@\textbf{e}_{3}@@, @@\textbf{e}_{4} \}@@: standard ordered basis (표준 기저)

e1=matrix(4,1,[1,0,0,0])

e2=matrix(4,1,[0,1,0,0])

e3=matrix(4,1,[0,0,1,0])

e4=matrix(4,1,[0,0,0,1])


Define a matrix @@T(\textbf{e}_{i})@@.

( @@T(a@@, @@b@@, @@c@@, @@d) [ i ]@@: (@@i+1@@)-th component of @@T(a@@, @@b@@, @@c@@, @@d)@@)

M=matrix(3,1,[T(1,0,0,0)[0], T(1,0,0,0)[1], T(1,0,0,0)[2]]);

M1=matrix(3,1,[T(0,1,0,0)[0], T(0,1,0,0)[1], T(0,1,0,0)[2]]);

M2=matrix(3,1,[T(0,0,1,0)[0], T(0,0,1,0)[1], T(0,0,1,0)[2]]);

M3=matrix(3,1,[T(0,0,0,1)[0], T(0,0,0,1)[1], T(0,0,0,1)[2]]);

M, M1, M2, M3

(
[1]   [2]   [ 0]   [0]
[1]   [0]   [-3]   [1]
[0],  [2],  [ 3],  [4]
)


Find the Matrix Representation of @@T@@  (선형변환 @@T@@의 행렬표현 구하기)

  (associated matrix, @@[[T(\textbf{e}_{1})]:[T(\textbf{e}_{2})]:[T(\textbf{e}_{3})]:[T(\textbf{e}_{4})]]@@)

M=M.augment(M1)   # 3 by 2 행렬

M=M.augment(M2)  # 3 by 3 행렬

M=M.augment(M3); # 3 by 4 행렬

M

[ 1  2  0  0 ]
[ 1  0 -3  1 ]
[ 0  2  3  4 ]


Check the result   (@@T \textbf{x}=M \textbf{x}@@임을 확인)

X=matrix(4,1,[x,y,z,w])

M*X

[         x + 2*y         ]
[   w + x           - 3*z]
[4*w +      2*y  + 3*z]



Find the Matrix Representation @@[T]_{\alpha}^{\beta}@@ with respected to standard ordered bases @@\alpha@@ and ordered bases @@\beta = \{ \textbf{v}_1@@, @@\textbf{v}_2@@, @@\textbf{v}_3 \}@@, where @@\textbf{v}_1@@, @@\textbf{v}_2@@, @@\textbf{v}_3@@ are as follows.


Define vectors. (벡터 생성)

v1=matrix(3,1,[1,1,0])

v2=matrix(3,1,[0,1,1])

v3=matrix(3,1,[1,0,1])


Declare variables. (변수 선언)

var('a,b,c')


To find matrix representation @@[T( \textbf{e}_i ) ]_{\alpha}^{\beta}@@,

                we have to solve the LSE @@T( \textbf{e}_i )=a_i \textbf{v}_1 + b_i \textbf{v}_2 + c_i \textbf{v}_3@@.




To find @@[T( \textbf{e}_1 ) ]_{\alpha}^{\beta}@@


Make equations (eq@@i@@ is @@i@@-th component of @@T( \textbf{e}_1 )-(a_1 \textbf{v}_1 + b_1 \textbf{v}_2 + c_1 \textbf{v}_3)@@)

eq1=(M*e1-(a*v1+b*v2+c*v3))[0,0]

eq2=(M*e1-(a*v1+b*v2+c*v3))[1,0]

eq3=(M*e1-(a*v1+b*v2+c*v3))[2,0]

eq1, eq2, eq3

(-1-c+1, -a-b+1, -b-c)


Solve the LSE. (연립 방정식 풀기)

solve([eq1==0, eq2==0, eq3==0], a,b,c)

[[a==1, b==0, c==0]]


@@MM=[[T(\textbf{e}_1)]_{\alpha}^{\beta}]@@

MM=matrix(QQ,3,1,[1,0,0]);

MM

[1]
[0]
[0]



To find @@[T( \textbf{e}_2 ) ]_{\alpha}^{\beta}@@


Make equations (eq@@i@@ is @@i@@-th component of @@T( \textbf{e}_2 )-(a_2 \textbf{v}_1 + b_2 \textbf{v}_2 + c_2 \textbf{v}_3)@@)

eq1=(M*e2-(a*v1+b*v2+c*v3))[0,0]

eq2=(M*e2-(a*v1+b*v2+c*v3))[1,0]

eq3=(M*e2-(a*v1+b*v2+c*v3))[2,0]

eq1, eq2, eq3

(-a-c+2, -a-b, -b-c+2)


Solve the LSE. (연립 방정식 풀기)

solve([eq1==0, eq2==0, eq3==0], a,b,c)

[[a==0, b==0, c==2]]


@@MM1=[[T(\textbf{e}_2)]_{\alpha}^{\beta}]@@

MM1=matrix(3,1,[0,0,2]);

MM1

[0]
[0]
[
2]



To find @@[T( \textbf{e}_3 ) ]_{\alpha}^{\beta}@@


Make equations (eq@@i@@ is @@i@@-th component of @@T( \textbf{e}_3 )-(a_3 \textbf{v}_1 + b_3 \textbf{v}_2 + c_3 \textbf{v}_3)@@

eq1=(M*e3-(a*v1+b*v2+c*v3))[0,0]

eq2=(M*e3-(a*v1+b*v2+c*v3))[1,0]

eq3=(M*e3-(a*v1+b*v2+c*v3))[2,0]

eq1, eq2, eq3

(-a-c, -a-b-3, -b-c+3)


Solve the LSE. (연립 방정식 풀기)

solve([eq1==0, eq2==0, eq3==0], a,b,c)

[[a==-3, b==0, c==3]]


@@MM2=[[T(\textbf{e}_3)]_{\alpha}^{\beta}]@@

MM2=matrix(3,1,[-3,0,3]);

MM2

[-3]
[0]
[
3]



To find @@[T( \textbf{e}_4 ) ]_{\alpha}^{\beta}@@


Make equations (eq@@i@@ is @@i@@-th component of @@T( \textbf{e}_4 )-(a_4 \textbf{v}_1 + b_4 \textbf{v}_2 + c_4 \textbf{v}_3)@@

eq1=(M*e4-(a*v1+b*v2+c*v3))[0,0]

eq2=(M*e4-(a*v1+b*v2+c*v3))[1,0]

eq3=(M*e4-(a*v1+b*v2+c*v3))[2,0]

eq1, eq2, eq3

(-a-c, -a-b+1, -b-c+4)


Solve the LSE. (연립 방정식 풀기)

solve([eq1==0, eq2==0, eq3==0], a,b,c)

 [[a==(-3/2), b==(5/2), c==(3/2)]]


@@MM3=[[T(\textbf{e}_4)]_{\alpha}^{\beta}]@@

MM3=matrix(3,1,[(-3/2),(5/2),(3/2)]);

MM3

[-3/2]
[
5/2]
[
3/2]





<Final> Augment each vector @@MMi@@ to @@MM@@ successively to make a associated matrix.

(선형변환 @@T@@의 행렬표현 구하기)

 (Matrix Representation of @@T=[[T(\textbf{e}_{1})]:[T(\textbf{e}_{2})]:[T(\textbf{e}_{3})]:[T(\textbf{e}_{4})]]=[MM:MM1:MM2:MM3]@@)

MM=MM.augment(MM1)

MM=MM.augment(MM2)

MM=MM.augment(MM3);

MM

[ 1  0  -3 -3/2 ]
[ 0  0   0  5/2 ]
[ 0  2   3  3/2 ]