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Worksheet 1

Section 1.1

• Are all triangular systems echelon? If not, write down a triangular system that is not echelon.

Yes.

• Are all echelon systems triangular? If not, write down a echelon system that is not triangular.

No. Consider the system consisting of the single equation $0=0$.

Section 1.2

• Find all solutions to the linear sytem of equations given by Example 3 in Section 1.2 in the book. If you're tired of solving linear systems, just look up the answer in the book. $$\begin{align} 2x_1 - 3x_2 + 10x_3 &= -2\\ x_1-2x_2 +3x_3 &= -2 \\ -x_1 +3x_2 +x_3 &=4. \end{align}$$

$(x_1,x_2,x_3)=(2,2,0)+s_1(-11,-4,1)$.

• Explicitly write down 2 solutions. In other words, write down 2 points with actual numbers that solve the linear system.

$(2,2,0), (-9, -2, 1)$.

• Find all solutions the homogenous linear system given by $$\begin{align} 2x_1 - 3x_2 + 10x_3 &= 0\\ x_1-2x_2 +3x_3 &= 0 \\ -x_1 +3x_2 +x_3 &=0. \end{align}$$ This is pretty much the same linear system but with the right hand side replaced by zeros.

$(x_1,x_2,x_3)=s_1(-11,-4,1)$.

• Call the 2 solutions you found earlier $a$ and $b$. Does $a-b$ solve the homogenous linear system? The answer should be yes. Can you explain why?

Yes. See homegenous solutions section.