The linear combinations of $v = (1, 1, 0)$ and $w = (0, 1, 1)$ fill a plane in $\mathbb R^3$.

1. Find a vector that is not a combination of $v$ and $w$ (i.e. not on the plane).
2. Find a vector perpendicular to that plane

For the vectors $v = ( 3, 4)$ and $w = ( 4, 3)$ test the Cauchy-Schwarz inequality and find whether they satisfy or not.

True or false.

1. The vectors $b$ that are not in the column space $C(A)$ form a subspace.
2. If $C(A)$ contains only the zero vector, then $A$ is the zero matrix.
3. The column space of $2A$ equals the column space of $A$.
4. The column space of $A - I$ equals the column space of $A$.

Do you know him?