Linear Algebra

True or False. The Intersection of Bases is a Basis of the Intersection of Subspaces

Problem 253

Determine whether the following is true or false. If it is true, then give a proof. If it is false, then give a counterexample.

Let $W_1$ and $W_2$ be subspaces of the vector space $\R^n$.
If $B_1$ and $B_2$ are bases for $W_1$ and $W_2$, respectively, then $B_1\cap B_2$ is a basis of the subspace $W_1\cap W_2$.

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Solution.

The statement is false. We give a counterexample.
Let us consider the vector space $\R^2$, the plane.

Let $W_1$ and $W_2$ be $\R^2$ itself. Then they are subspaces of $\R^2$.
Let
\[B_1=\left\{\begin{bmatrix}
1 \\
0
\end{bmatrix}, \begin{bmatrix}
0 \\
1
\end{bmatrix} \right\}\text{ and }
B_2=\left\{ \begin{bmatrix}
-1 \\
0
\end{bmatrix}, \begin{bmatrix}
0 \\
-1
\end{bmatrix} \right\}\] be bases of $W_1$ and $W_2$, respectively.

(Note. $B_1$ consists of two linearly independent vectors in the 2-dimensional vector space $W_1=\R^2$, hence $B_1$ is a basis of $W_1$. Similarly, $B_2$ is a basis of $W_2$.)

Since $W_1$ and $W_2$ are both $\R^2$, we have $W_1\cap W_2=\R^2$.
However, the intersection $B_1\cap B_2$ is the empty set, and the empty set is not a basis of $W_1\cap W_2=\R^2$. Thus, we have found a counterexample to the statement.

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