If $l$ is an integer such that $k \le l \le m$, show that there exists subspace $X$ of $V$ such that $U \subseteq X \subseteq W$ and $\dim(X)=l$.

Question

Let $U$ and $W$ be subspaces of a vector space $V$ such that $U \subseteq W, \dim(U)=k,$ dan $\dim(W)=m$ with $k \lt m$.

If $l$ is an integer such that $k \le l \le m$, show that there exists subspace $X$ of $V$ such that $U \subseteq X \subseteq W$ and $\dim(X)=l$.

Any idea how to starting with? What theorem or properties or other in dimension of vector space could I use? Thanks in advanced.

Answer 1

Here are the basic ingredients:

• Every linearly independent set in a vector space can be extended to a basis (hint: $U$ is a subspace of $W$).
• Every subset of a linearly independent set is linearly independent, and is a basis for its span.

Let me know if you need more help!