Vector Space Isomorphisms Examples 1

Recall from the Vector Space Isomorphisms page that if $V$ and $W$ are vector spaces over the field $\mathbb{F}$ then $V$ and $W$ are said to be isomorphic to each other, written $V \cong W$ if there exists an invertible linear map $T$ between $V$ and $W$.

We saw that isomorphic finite-dimensional vector spaces have the same dimension, and more succinctly, two finite-dimensional vector spaces are isomorphic if and only if they have the same dimension.

We will now look at some examples regarding vector space isomorphisms.

Example 1

Let $U_1$, $U_2$, and $W$ be subspaces of the finite-dimensional vector space $V$ where $U_1 + W = U_2 + W$ and where $U_1 \cap W = U_2 \cap W$. Prove that then $U_1$ is isomorphic to $U_2$.

To show that $U_1$ is isomorphic to $U_2$, we want to show that $\mathrm{dim} (U_1) = \mathrm{dim} (U_2)$.

Since $V$ is a finite-dimensional vector space then the subspaces $U_1$, $U_2$, and $W$ are all finite-dimensional. We will apply the dimension formula for the sum of subspaces to get the following two equalities:

(1)

\begin{align} \quad \mathrm{dim} (U_1 + W) = \mathrm{dim} (U_1) + \mathrm{dim} (W) - \mathrm{dim} (U_1 \cap W) \end{align}

(2)

\begin{align} \quad \mathrm{dim} (U_2 + W) = \mathrm{dim} (U_2) + \mathrm{dim} (W) - \mathrm{dim} (U_2 \cap W) \end{align}

Since $U_1 + W = U_2 + W$ we have that $\mathrm{dim} (U_1 + W) = \mathrm{dim} (U_2 + W)$ and so:

(3)

\begin{align} \quad \mathrm{dim} (U_1) + \mathrm{dim} (W) - \mathrm{dim} (U_1 \cap W) = \mathrm{dim} (U_2) + \mathrm{dim} (W) - \mathrm{dim} (U_2 \cap W) \\ \quad \mathrm{dim} (U_1) - \mathrm{dim} (U_1 \cap W) = \mathrm{dim}(U_2) - \mathrm{dim} (U_2 \cap W) \end{align}

Since $U_1 \cap W = U_2 \cap W$ we have that $\mathrm{dim} (U_1 \cap W) = \mathrm{dim} (U_2 \cap W)$ and so this implies that:

(4)

\begin{align} \quad \mathrm{dim} (U_1) = \mathrm{dim} (U_2) \end{align}

Therefore $U_1$ is isomorphic to $U_2$.