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**1. Homework Statement**

In R^3 with inner product calculate all the least square solutions, and choose the one with shorter length, of the system:

x + y + z = 1

x + z = 0

y = 0

**2. The attempt at a solution**

So I applied the formula A^T A x = A^T b with A as being the matrix with row 1 (1,1,1) row 2 (1,0,1) and row 3 (0,1,0); x being the column (x_1,x_2,x_3) and b being the column (1,0,0).

So I did it and I reached to the solution (x_1, \frac {1}{3}, \frac {1}{3} + x_1)

And I expanded this solution in two vectors (0, \frac {1}{3}, \frac {1}{3}) and (1,0,1).

So these are the least square solutions and the one with shorter length is the first one.

My doubt is if I'm doing this correctly or if I made any mistake because I used an online calculator that only give one least square solution. Can someone help me to verify my attempt? Thanks!