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Linear Algebra with Sage

<Reduced Row Echelon Form>


Made by SKKU Linear Algebra Lab (2011)



Solve following LSE using RREF.

@@ \left\{ \begin{array}{ll} y+2z=9\\ 5x+10y-15z=1\\ 3x+6y-8z=0 \end{array}\right.@@



<There are 3 cases : Unique solution, Infinitely many solutions, No solution>

(?쇰컲?곸쑝濡?二쇱뼱吏??됰젹??REF ?먮뒗 RREF 瑜?援ы븳 ?? ?꾩쓽 3媛€吏€ 寃쎌슦 以??대뒓 寃쎌슦?몄?瑜??쎄쾶 ?뚯븙???? 寃쎌슦???곕씪 ?닿껐?섎㈃ ?쒕떎.)


First, we do find REF or RREF of a given matrix.

Read: http://matrix.skku.ac.kr/2008-Lecture/LAA-sglee/text-ch01SPF/Page020.html


Define matrices (怨꾩닔?됰젹怨??곸닔?됰젹 ?앹꽦 諛??뺤씤)

A=matrix(QQ,[[0,1,2],[5,10,-15],[3,6,-8]])

v=matrix(3,1,[9,1,0])

print A

print v

[  0   1   2]

[  5  10 -15]

[  3   6  -8]

[9]

[1]

[0]


Make a augmented matrix (泥④??됰젹 ?앹꽦)

Av=A.augment(v);Av

[  0   1   2   9]

[  5  10 -15   1]

[  3   6  -8   0]


ERO: @@R_1 \leftrightarrow R_2@@

Av.swap_rows(0,1);Av # 1?됯낵 2?됱쓣 諛붽씔??

[  5  10 -15   1]

[  0   1   2   9]

[  3   6  -8   0]


ERO: @@\frac{1}{5} R_1 \rightarrow R_1@@

Av.rescale_row(0,1/5);Av # 1?됱뿉 1/5諛곕? ?쒕떎.

[  1   2  -3 1/5]

[  0   1   2   9]

[  3   6  -8   0]


ERO: @@R_3 - 3 R_1 \rightarrow R_3@@

Av.add_multiple_of_row(2,0,-3);Av # 1?됱뿉 -3諛곕? ???ㅼ쓬 3?됱뿉 ?뷀븳??

[   1    2   -3  1/5]

[   0    1    2    9]

[   0    0    1 -3/5]


Find an echelon form of matrix (REF ?먮뒗 RREF 援ы븯湲?

Av.echelon_form() # 二쇱뼱吏??됰젹 Av媛€ ZZ濡??뺤쓽?섎㈃ REF瑜?蹂댁뿬二쇨퀬, QQ濡??뺤쓽?섎㈃ RREF瑜?蹂댁뿬以€??

[   1    0    0  -22]

[   0    1    0 51/5]

[   0    0    1 -3/5]


Find a RREF of matrix (RREF 援ы븯湲?

Av.rref() # 二쇱뼱吏??됰젹 Av???뺤쓽(ZZ, QQ)?€???곴??놁씠 RREF瑜?蹂댁뿬以€??

[   1    0    0  -22]

[   0    1    0 51/5]

[   0    0    1 -3/5]



Then, If Case 1(Unique solution): RREF@@(A)=I@@,


Solve the LSE using inverse matrix (紐낅졊?대? ?댁슜??LSE????援ы븯湲?

A\v # A.inverse()*v

[ -22]

[51/5]

[-3/5]



If Case 2 and 3(Infinitely many solutions, No solution): RREF@@(A)\neq I@@,


Read: http://matrix.skku.ac.kr/2008-Lecture/LAA-sglee/text-ch01SPF/Page023.html