Linear Algebra and Its Applications, Exercise 2.3.17

Exercise 2.3.17. Suppose that V and W are subspaces of \mathbf{R}^5, each with dimension 3. Show that V and W must have at least one vector in common other than the zero vector.

Answer: Since V and W each have dimension 3 their respective bases each contain three vectors. Let v_1, v_2, and v_3 be a basis for V and w_1, w_2, and w_3 be a basis for W.

Now consider the combined set of six vectors. Since we have six vectors in a vector space of dimension 5 the combined set of vectors is linearly dependent, with at least one vector expressible as a linear combination of the other five vectors. Without loss of generality assume that w_3 is dependent on the other five vectors, so that we have

w_3 = c_1v_1 + c_2v_2 + c_3v_3 + c_4w_1 + c_5w_2

for some set of weights c_1 through c_5. We can rearrange the above equation as follows:

c_1v_1 + c_2v_2 + c_3v_3 = -c_4w_1 - c_5w_2 + w_3

Now consider the vector u = c_1v_1 + c_2v_2 + c_3v_3. Since u is a linear combination of the basis vectors v_1, v_2, and v_3 it is in the subspace V. But from the above equation we also have u = -c_4w_1 - c_5w_2 + w_3 so that u is a linear combination of the basis vectors w_1, w_2, and w_3 and thus is also in the subspace W.

So u is a member of both V and W. Now suppose u = 0. We then have -c_4w_1 - c_5w_2 + w_3 = 0 or w_3 = c_4w_1 + c_5w_2 so that w_3 is a linear combination of w_1 and w_2 and the set of vectors w_1, w_2, and w_3 is linearly dependent. But this contradicts the assumption that w_1, w_2, and w_3 form a basis for W and are thus linearly independent. Since the assumption u = 0 leads to a contradiction we conclude that u \ne 0.

We have thus shown that there must exist a nonzero vector u that is a member of both V and W.

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.

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