Exercises on column space and nullspace
Problem 6.1: (3.1 #30. Introduction to Linear Algebra: Strang) Suppose S and T are two subspaces of a vector space V.
a) Definition: The sum S + T contains all sums s + t of a vector s in S and a vector t in T. Show that S + T satisfies the requirements (addition and scalar multiplication) for a vector space.
b) If S and T are lines in Rm, what is the difference between S + T and S ∪ T? That union contains all vectors from S and T or both. Explain this statement: The span of S ∪ T is S + T.
Problem 6.2: (3.2 #18.) The plane x − 3y − z = 12 is parallel to the plane x − 3y − x = 0. One particular point on this plane is (12, 0, 0). All points on the plane have the form (fill in the first components)
.
Problem 6.3: (3.2 #36.) How is the nullspace N(C) related to the spaces
N(A) and N(B), if C ?
