Find a basis for the set of vectors in R3 in the plane x 2y

Find a basis for the set of vectors in R^3 in the plane x + 2y - 3z = 0. (b) Find a basis for the set of vectors in R^4 in the subspace (hyperplane) x_1 + x_3 + 2x_4 = 0. x_1 + 2x_2 - x_3 + x_4 = 0.

Solution

the reduced echleon form is

which corresponds to the system

The leading entries in the matrix have been highlighted in yellow.

A leading entry on the (i,j) position indicates that the j-th unknown will be determined using the i-th equation.

Those columns in the coefficient part of the matrix that do not contain leading entries, correspond to unknowns that will be arbitrary.

The system has infinitely many solutions:

The solution can be written in the vector form:

c₃ +

c₄

1	0	1	2
1	2	-1	1