질문) 왜 두 rows (row 1 and 2) 의 (linear) combination 으로 생성된 row 3 가 생성하는 평면은 row1, row2 와 삼각형 터널을 이루는거임? (어떤 그림인지 상상은 간다만) 왜 그래야하는지 잘 모르겠음..



질문의 배경: 3 by 3 연립방정식 문제 (Ax = b)

문제)
Three planes can fail to have an intersection point, even if no planes are parallel.

i) The system is singular if row 3 of A is a ____ of the first two rows.

ii) Find a third equation that can't be solved together with x + y + z = 0 and x - 2y - z = l.


솔루션)
Singular if row 3 is a combination of rows 1 and 2. From the end view, the three planes form a triangle.

This happens if rows 1+2=row 3 on the left side but not the right side: x+y+z=0, x−2y−z=1, 2x−y=4.

No parallel planes but still no solution.

The three planes in the row picture form a triangular tunnel.