Description
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)
⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ x
⎣ y ⎦ = ⎣ 0 ⎦ + y ⎣ 1 ⎦ + z ⎣ 0 ⎦ .
z 0 0 1
Problem 6.3: (3.2 #36.) How is the nullspace N(C) related to the spaces
A N(A) and N(B), if C = ? B
1
MIT OpenCourseWare
https://ocw.mit.edu
18.06SC Linear Algebra
Fall 2011
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