For subsets $M\subset N$ of a vector space, $\langle{M\rangle}\subset \langle{N\rangle}$.

Question

Hello I had to prove that $M\subset N$ implies $\langle{M}\rangle \subset \langle{M}\rangle$ for $M, N$ subsets of a vector space. My idea looks like the following: Suppose $ M = \{v_1, ..., v_m\} $ and $ N = \{v_1, ..., v_n\} $ $with \space 1 \leq m \leq n \space m,n \in N$ From that it follows by applying the definition of a linear hull: $$〈M〉=K \cdot v_1 + ... + K \cdot v_m$$ and $$〈N〉=K \cdot v_1 + ... + K \cdot v_n$$ Because m<=n it follows that 〈M〉⊂〈N〉

Apparently that was the wrong way to do it though as i assumed that the sets are finite. Can somebody help me out here? How do you proof this correctly? Thanks in advance!

Answer 1

For a vector space $V$ and a subset $X\subset V$, the space $\langle{X}\rangle$ is by definition or construction the minimal subspace of $V$ containing $X$. But $\langle{N\rangle}$ then also contains $M\subset N$, and so it must also contain $\langle{M\rangle}$.

Answer 2

Since M $\subset$ N, every vector of M is also a vector of N. So a linear combination of elements of M can be viewed as a linear combination of elements in N. Thus $<M> \subset<N>.$