Basis for a plane If I have a given plane, and I choose two linearly independent vectors within the plane, why must they be a basis for the plane? I'm having trouble conceptualizing this. 
 A: A plane is fundamentally built out of two directions. They are ideally orthogonal but this is not necessary. Any pair of different directions within that plane are sufficient to form all of it (i.e. the vectors representing those directions form a basis).
Two vectors that are linearly independent are by definition different directions, and if both come from the plane of interest, they are enough to build exactly that plane, no more and no less.
Hence, any LI pair of vectors are a basis for a plane, and any LI pair of vectors from a specific plane are a basis for that plane (and no other).
A: Take a flat sheet of paper. Take any point as the origin. Draw any 2 non-collinear vectors starting at the origin. Draw a third vector. Is it possible that this vector is not the sum of the two other vectors eventually multiplied by scalars (positive or negative)?
A: A plane , if it passes through the origin, has dimension 2. So any two linearly independent vectors in the plane must be a basis.
A: Define the plane as $2$ linearly independent vectors such that
$$v = \begin{bmatrix}a\\0\end{bmatrix}, u = \begin{bmatrix}0\\b\end{bmatrix}$$
and $a, b \in \mathbb{R}$
$$v + u = \begin{bmatrix}a\\b\end{bmatrix}$$
Now that we have added the vectors, can you see how we can get to every point on the plane now by scaling $a$ and $b$?
