Vector basis calculator

1. How do you know if vectors form a basis for R3?
2. How do you find the basis of a subspace?
3. How do you find the basis of a matrix?

How do you know if vectors form a basis for R3?

The set has 3 elements. Hence, it is a basis if and only if the vectors are independent. Since each column contains a pivot, the three vectors are independent. Hence, this is a basis of R3.

How do you find the basis of a subspace?

If you want to find a basis for S=Span(v1,v2,v3,v4) you can write the vectors as rows of a 4×4 matrix, do row reduction, and when you are done, the non-zero rows are a basis for S (this is because row reduction does not change the row space).

How do you find the basis of a matrix?

In order to compute a basis for the null space of a matrix, one has to find the parametric vector form of the solutions of the homogeneous equation Ax = 0.