Vector Subspace Question (GRE 0568 Q3)

This is an interesting question on vector subspaces (a topic from linear algebra):

If V and W are 2-dimensional subspaces of \mathbb{R}^4, what are the possible dimensions of the subspace V\cap W?

(A) 1 only
(B) 2 only
(C) 0 and 1 only
(D) 0, 1, and 2 only
(E) 0, 1, 2, 3, and 4

To begin this question, we would need this theorem on the dimension of sum and intersection of subspaces (for finite dimensional subspaces):

\dim (M+N)=\dim M+\dim N-\dim (M\cap N)

Note that this looks familiar to the Inclusion-Exclusion principle, which is indeed used in the proof.

Hence, we have \dim(M\cap N)=\dim M+\dim N-\dim (M+N)=4-\dim (M+N).

\dim (M+N), the sum of the subspaces M and N, is at most 4, and at least 2.

Thus, \dim (M\cap N) can take the values of 0, 1, or 2.

Answer: Option D

If you are looking for a lighthearted introduction on linear algebra, do check out Linear Algebra For Dummies. Like all “For Dummies” book, it is not overly abstract, rather it presents Linear Algebra in a fun way that is accessible to anyone with just a high school math background. Linear Algebra is highly useful, and it is the tool that Larry Page and Sergey Brin used to make Google, one of the most successful companies on the planet.

Author: mathtuition88

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

This site uses Akismet to reduce spam. Learn how your comment data is processed.

%d bloggers like this: