## 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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