Suppose we are asked if $ \{\begin{bmatrix}a\\b\end{bmatrix}, \begin{bmatrix}c\\d\end{bmatrix}\}$ is a basis for Span $ \{\begin{bmatrix}e\\f\end{bmatrix}, \begin{bmatrix}g\\h\end{bmatrix}, \begin{bmatrix}i\\j\end{bmatrix}\}$ where $a, b, c, d, e, f, i, j$ are all scalars.
Here's my approach so far: Firstly, we check if $ \begin{bmatrix}a\\b\end{bmatrix}$ and $\begin{bmatrix}c\\d\end{bmatrix}$ are linearly independent by row reducing the vectors in $ \{\begin{bmatrix}a\\b\end{bmatrix}, \begin{bmatrix}c\\d\end{bmatrix}\}$ and checking if we have a pivot in every column. Then, we check if Span $ \{\begin{bmatrix}a\\b\end{bmatrix}, \begin{bmatrix}c\\d\end{bmatrix}\}$ = Span $ \{\begin{bmatrix}e\\f\end{bmatrix}, \begin{bmatrix}g\\h\end{bmatrix}, \begin{bmatrix}i\\j\end{bmatrix}\}$ by checking if Span $ \{\begin{bmatrix}a\\b\end{bmatrix}, \begin{bmatrix}c\\d\end{bmatrix}\}$ $\subseteq$ Span $ \{\begin{bmatrix}e\\f\end{bmatrix}, \begin{bmatrix}g\\h\end{bmatrix}, \begin{bmatrix}i\\j\end{bmatrix}\}$ and vice versa. To check if Span $ \{\begin{bmatrix}a\\b\end{bmatrix}, \begin{bmatrix}c\\d\end{bmatrix}\}$ $\subseteq$ Span $ \{\begin{bmatrix}e\\f\end{bmatrix}, \begin{bmatrix}g\\h\end{bmatrix}, \begin{bmatrix}i\\j\end{bmatrix}\}$, I first pick $ \begin{bmatrix}a\\b\end{bmatrix}$ and see if $\begin{bmatrix}e&g&i&a\\f&h&j&b\end{bmatrix}$ is consistent. Then, I pick $ \begin{bmatrix}c\\d\end{bmatrix}$ and see if $\begin{bmatrix}e&g&i&c\\f&h&j&d\end{bmatrix}$ is consistent. I repeat the same process to check if Span $ \{\begin{bmatrix}e\\f\end{bmatrix}, \begin{bmatrix}g\\h\end{bmatrix}, \begin{bmatrix}i\\j\end{bmatrix}\}$ $\subseteq$ Span $ \{\begin{bmatrix}a\\b\end{bmatrix}, \begin{bmatrix}c\\d\end{bmatrix}\}$.
Is my approach accurate?
Also, on a somewhat related note, if I have determined that $ \{\begin{bmatrix}e\\f\end{bmatrix}, \begin{bmatrix}g\\h\end{bmatrix}\}$ is a basis for Span $ \{\begin{bmatrix}e\\f\end{bmatrix}, \begin{bmatrix}g\\h\end{bmatrix}, \begin{bmatrix}i\\j\end{bmatrix}\}$, then is it necessarily true that say $ \{\begin{bmatrix}e\\f\end{bmatrix}\}$ cannot be a basis for Span $ \{\begin{bmatrix}e\\f\end{bmatrix}, \begin{bmatrix}g\\h\end{bmatrix}, \begin{bmatrix}i\\j\end{bmatrix}\}$?
Best Answer
This looks correct on the whole, but it’s way too much work. For one thing, you’re unnecessarily redoing some of the work: you can check all of the vectors in one set against the span of the other set at once by assembling them all into a single matrix and row-reducing that instead of doing this one-by-one. You could also skip checking if the set of two vectors is linearly independent, since that information is only relevant once you know the dimension of the span of the set of three. So, just dive right into row-reducing the matrix $$\left[\begin{array}{ccc|cc} e&g&i & a&c \\ f&h&j & b&d\end{array}\right].$$ Assuming that neither set of vectors consists entirely of the zero vector (in which case you can trivially tell if the spans are the same) there are two cases to consider. If the bottom row of the left-hand side is zero, then the spans are the same if the entire row is zero: this is you “consistent equations” test. If the bottom row of the left-hand size is nonzero, then the right-hand side must also have rank 2 for the spans to match. This is where you need to know about the linear independence of the pair of vectors on the right. Any set of two vectors is linearly dependent iff one is a scalar multiple of the other, which you should be able to tell at a glance, or at worst, by comparing ratios of corresponding components.
You could even start with that test: if the pair of vectors is linearly-dependent and at least one is nonzero, then for the spans to match every vector in the set of three must also be a scalar multiple of this nonzero vector. If the pair is linearly independent, then since you’re working in $\mathbb R^2$, it spans the entire space, so you only need to check the dimension of the other span, which you can do by row-reducing the matrix $$\begin{bmatrix}e&g&i\\f&h&j\end{bmatrix}$$ or, equivalently, seeing if $(e,g,i)$ is a scalar multiple of $(f,h,j)$ or vice-versa.
As for the second question, a single vector can’t span a two-dimensional space.