Just two quick questions.
i) When solving a linear system and you've reduced the matrix to reduced row echelon form. For example:
1 -2 0 5 0
0 0 1 2 0
0 0 0 0 1
0 0 0 0 0
How do you determine what the free variables are in this equation or in any equation in general? Is it always the ones that don't have coefficients of 1?
ii) let a b c be scalars such that abc != 0.
prove that ax + by + cz = 0 is a subspace of R3.
Now the conditions to be a subspace is such that the subset has to be closed under vector addition and scalar multiplication.
now the book says that ax+by+cz is a solution to the homogenous system Ax = 0
In this case A would be the row vector a, b, c. and x would be the column vector
x, y ,z.
Now I'm a bit confused as to how you can describe the solution as a plane of ax + by + cz and how can you use that to prove that its a subspace of R3.