Linear Algebra and Its Applications, Exercise 2.3.12

Exercise 2.3.12.Suppose that the set of vectors v_1, v_2, v_3,  and v_4,  is a basis for \mathbf{R}^4 and that W is a subspace of \mathbf{R}^4. Provide a counterexample to the conjecture that some subset of v_1, v_2, v_3,  and v_4 is necessarily a basis for W.

Answer: Suppose that v_1, v_2, v_3,  and v_4 are equal to the vectors (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), and (0, 0, 0, 1) and suppose that W is the subspace consisting of all vectors whose first two elements are equal to each other and whose last two elements are equal to each other; i.e., vectors in W are of the form (a, a, c, c). (It is fairly simple to verify that W is in fact a subspace, so I omit that here.) In this case none of the vectors v_1, v_2, v_3,  and v_4 are in the subspace W and thus cannot be part of a basis for W.

Instead we could use, for example, the vectors (1, 1, 0, 0) and (0, 0, 1, 1) as a basis for the subspace W, with any vector w = (a, a, c, c) in the subspace expressible as

w = a \begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix} + c \begin{bmatrix} 0 \\ 0 \\ 1 \\ 1 \end{bmatrix}

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.

This entry was posted in linear algebra. Bookmark the permalink.

Leave a Reply

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

WordPress.com Logo

You are commenting using your WordPress.com 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 )

Google+ photo

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

Connecting to %s