Can a vector in $\mathbb R^2$ be a subspace of $\mathbb R^3$?

Question

Is it possible for a set of vectors that don't span $\mathbb R^3$ to be the subspace of $\mathbb R^3$?

Given the vector space $V = \{[a-b+c, a+b-c, 0]\}$, if we make a set of vectors out of this vector space then it would look something like this $\{[1, 1, 0], [-1, 1, 0], [1, -1, 0]\}$. Now, if make a matrix of these vectors and then reduce that in the row echelon form, it will have two pivot entries and according to invertible matrix theorem, the set of vectors spans $\mathbb R^2$.

Moreover, according to one of the theorems in linear algebra "The column space of a $m \times n$ matrix is a subspace of $\mathbb R^m$" which implies that the set of the above-given vectors is the subspace of $\mathbb R^3$.

This seems to be confusing to me. If the set of vectors doesn't span $\mathbb R^3$ then how can it be a subspace of $\mathbb R^3$?

Answer 1

Note that in $\mathbb{R^3}$ we can have the following proper subspaces

• the origin
• lines through the origin
• planes through the origin (that's the case of V in your example)

and each of them obviously doesn't span $\mathbb{R^3}$.