## 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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