Linear Algebra 2.8 Definitions

Closed under addition and scalar multiplication.

• Rn is a subspace of itself.
• Zero subspace is a subspace of Rn--set consisting of only the zero vector in Rn.

• The zero vector is in H
• For each u and v in H, the sum u + v is in H
• For each u in H and each scalar c, the vector cu is in H.

the set of Col A of all linear combinations of the columns of A

pivot columns of A (not row reduced) form basis for column space of A.

set of Nul A of all solutions of the homogeneous equation Ax = 0

• Nul A is a subspace of Rn.
• The set of solutions of a system of Ax = 0 is also a subspace of Rn.

a linearly independent set in H that spans H

Row reduce Ax=0, put into vector equation form, and the vectors are the basis

rows with pivots in row reduced matrix