This is the solution to one of the problems from my linear algebra quiz. I under

Question

This is the solution to one of the problems from my linear algebra quiz. I understand how to solve for x,y, and z through row operations , but I have no idea what happens in the last 2 steps; like how you express your answer as a span of a set of vectors. If someone could explain how you get there that'd be great

solve the following system and express your answer in terms of a span. x+2y-3z=5 2x+y-3z=13 -x+y=-8 [1 2 -3 2 1 -3 -1 1 0|5 13 -8]2R1-R2 rightarrow R2 R1+R3 rightarrow R3 [1 2 -3 0 3 -3 0 3 -3|5 -3 -3] R2-R3 rightarrow R3 1/3R2 rightarrow R2[1 2 -3 0 1 -1 0 0 0|5 -1 0]R1-2R2 rightarrow R2 [1 0 -1 0 1 -1 0 0 0|7 -1 0] last row indicates that the system is consistent. x-z=7 y-z=-1 x=z+7 y=z-1 Rewrite the solution of a vector [x y z]=[z+7 z-7 z]=[1 1 1]z+[7 -1 0] The solution set is span([1 1 1])+(7 -1 0)

Explanation / Answer

In the 2nd last step, you are just splitting the matrix as a sum of two other matrices as "z" is common in all the three rows.

Once that is done, you are writing the solution as the spannin] set (1,1,1) + (7,-1,0) [ as z is a variable and for any value of z the matrix z*(1,1,1) + (7,-1,1) would give you the solution)

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