State the linear-combination test for a subset \(W\) of a vector space \(V\) to be a subspace.

Why must every subspace contain the zero vector?

Use the zero-vector test to decide whether \(A=\{(x,y)\in\mathbb R^2:y=3\}\) is a subspace.

Is the line \(L=\{t(2,-1):t\in\mathbb R\}\) a subspace of \(\mathbb R^2\)?

Show that \(W=\{(x,y,z)\in\mathbb R^3:z=0\}\) is closed under addition.

Show that \(W=\{(x,y,z)\in\mathbb R^3:z=0\}\) is closed under scalar multiplication.

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

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

Show that \(W=\{(x,y,z):x-2y+z=0\}\) is a subspace of \(\mathbb R^3\).

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

Let \(U=\{(x,y,z):x+y=0\}\) and \(W=\{(x,y,z):z=0\}\). Show that \(U\cap W\) is a subspace.

Explain why \(C=\{(x,y)\in\mathbb R^2:x\ge0\}\) is not a subspace even though it contains \((0,0)\).

Let \(P_2\) be polynomials of degree at most \(2\). Show that \(W=\{p\in P_2:p(0)=0\}\) is a subspace.

Let \(M\) be all real \(2\times2\) matrices. Is \(W=\{A\in M:A^T=A\}\) a subspace?

For which real \(c\) is \(W_c=\{(x,y,z):x+y+z=c\}\) a subspace of \(\mathbb R^3\)?

For which real \(k\) is \(L_k=\{(x,y):y=kx+2\}\) a subspace?

For which real \(a\) is \(W_a=\{(x,y):y=ax\}\) a subspace of \(\mathbb R^2\)?

A student says the union of two subspaces is always a subspace. Give a counterexample in \(\mathbb R^2\).

Prove that the intersection of two subspaces \(U\) and \(W\) of \(V\) is a subspace of \(V\).

A subset \(W\subseteq V\) is non-empty and closed under all linear combinations \(\lambda\mathbf u+\mu\mathbf v\). Prove that \(W\) is a subspace.