## Related questions with answers

For the given problem, consider the homogeneous systems of linear equations. For convenience, general solutions are given. It can be shown, as in the previous exercise, that the sets of solutions form subspaces of $\mathrm{R}^4$.

(a) Use the general solution to construct two specific solutions. (b) Use the operations of addition and scalar multiplication to generate four vectors from these two solutions. (c) Use the general solution to check that these vectors are indeed solutions, giving the values of $r$ and $s$ for which they are solutions.

$\begin{aligned} x_1+x_2+x_3+3 x_4&=0\\ x_1+2 x_2+4 x_3+5 x_4&=0\\ x_1 \quad-2 x_3+x_4&=0 \end{aligned}$

General solution is $(2 r-s,-3 r-2 s, r, s)$.

Solution

VerifiedFor the system

$\begin{alignedat}{4} x_1 \; &+& \; x_2 \; &+& \; x_3 \; &+&\; 3x_4&=0\\ x_1 \; &+& \; 2x_2 \; &+& \; 4x_3 \; &+&\; 5x_4&=0\\ 3x_1 && &-& \; 2x_3 \; &+&\; x_4&=0\\ \end{alignedat}$

we are given the general solution $(2r-s,-3r-2s,r,s)$.

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