MATH 214
Linear Algebra
Week 8
Homework (Due 3/24)
Date Section Assignment
3/7 4.1 1, 2, 6, 9, 12, 21, 24, 24, 26, 28, 31
Bonus Question
1. Let Hand Kbe subspaces of a vector space V. The intersection of Hand K, written as
H∩K, is the set of vectors that belong to both Hand K. Show that H∩Kis a subspace
of V. Give an example in R2to show that the union of two subspaces is not, in general, a
subspace.
Proof. We have that 0∈Hbecause His a subspace of V, and 0∈Kbecause Kis a subspace
of V. Hence 0∈H∩K.
Suppose u,v∈H∩K. Then u,v∈H. Hence u+v∈Hbecause His a subspace of V.
Similarly, u,v∈K. Hence u+v∈Kbecause Kis a subspace of V. Thus u+v∈H∩K.
Suppose u∈H∩Kand cis a scalar. Then u∈Hand u∈K. Hence cu∈Hand cu∈K
because Hand Kare subspaces of V. Thus cu∈H∩K.
Therefore H∩Kis a subspace of V.
To show that the union of two subspaces in not necessarily a subspace, let H= Span 1
0
and K= Span 0
1. Then 1
0∈H∪Kand 0
1∈H∪K, but
1
0+0
1=1
1/∈H∪K.
Other counterexamples will work, but Hand Kmust be subspaces of R2, and you must find
specific vectors uand vsuch that u,v∈H∪Kand u+v/∈H∪K. Is it possible to find a
vector uand a scalar csuch that u∈H∪Kbut cu/∈H∪K?
1