Explain why {\((1, 0)\), \((0, 1)\)} is a basis for \(\mathbb R^2\).

Is {\((1, 2)\)} a basis for \(\mathbb R^2\)? Explain.

Is {\((1, 0)\), \((0, 1)\), \((1, 1)\)} a basis for \(\mathbb R^2\)? Explain.

Use a determinant to decide whether {\((1, 2)\), \((3, 5)\)} is a basis for \(\mathbb R^2\).

Find a basis for the line \(L=span\)(\((2, 4, 6)\)).

Find a basis for the plane x + y + \(z=0\) in \(\mathbb R^3\).

Is {\((1, 0, 0)\), \((0, 1, 0)\)} a basis for the xy-plane in \(\mathbb R^3\)?

Find a basis from the spanning set {\((1, 0, 0)\), \((0, 1, 0)\), \((1, 1, 0)\)} for the xy-plane.

Can two vectors be a basis for \(\mathbb R^3\)? Explain.

A learner says a basis is any spanning set. Diagnose the error.