Decide whether S = {\((x, y)\) in \(\mathbb R^2\) : y = 2x} is a subspace of \(\mathbb R^2\).

Decide whether S = {\((x, y)\) in \(\mathbb R^2\) : \(y=x\) + 1} is a subspace of \(\mathbb R^2\).

Decide whether S = {\((x, y, z)\) in \(\mathbb R^3\) : x + y + \(z=0\)} is a subspace of \(\mathbb R^3\).

Decide whether S = {\((x, y)\) in \(\mathbb R^2\) : x >= 0} is a subspace of \(\mathbb R^2\).

Decide whether the xy-plane, S = {\((x, y, z)\) in \(\mathbb R^3\) : \(z=0\)}, is a subspace of \(\mathbb R^3\).

Decide whether S = {\((x, y, z)\) in \(\mathbb R^3\) : \(z=1\)} is a subspace of \(\mathbb R^3\).

Let \(S=span\)(\((1, 2, 0)\), \((0, 1, 3)\)). Explain why S is a subspace of \(\mathbb R^3\).

Show that S = {\((x, y, z)\) in \(\mathbb R^3\) : \(x=0\) and \(y=0\)} is a subspace and describe it geometrically.

Decide whether S = {\((x, y)\) in \(\mathbb R^2\) : \(xy=0\)} is a subspace of \(\mathbb R^2\).

A learner says any line in \(\mathbb R^2\) is a subspace. Diagnose the error.