State the two closure properties a real vector space must satisfy.

What is the zero vector in \(\mathbb R^3\), and why must it belong to any vector space structure on \(\mathbb R^3\) with usual addition?

Let \(V=\{(x,y)\in\mathbb R^2:x\ge 0\}\) with usual operations. Use one scalar multiple to decide whether \(V\) is a vector space.

Let \(S\) be the set of unit vectors in \(\mathbb R^2\). Show that \(S\) is not a vector space using the vector \((1,0)\).

Check whether \(V=\{(x,y)\in\mathbb R^2:y=2x\}\) is closed under addition.

Check whether \(V=\{(x,y)\in\mathbb R^2:y=2x\}\) is closed under scalar multiplication.

Decide whether \(A=\{(x,y)\in\mathbb R^2:y=x+1\}\) is a vector space with usual operations.

Let \(P\) be the set of all real polynomials of degree exactly \(2\). Explain why \(P\) is not a vector space.

Show that the set \(V=\{(x,y,z)\in\mathbb R^3:x+y+z=0\}\) is closed under addition and scalar multiplication.

For real functions on \(\mathbb R\), let \(E\) be the set of even functions, where \(f(-x)=f(x)\). Show that \(E\) is closed under addition and scalar multiplication.

Decide whether the set \(O\) of odd real functions, satisfying \(f(-x)=-f(x)\), is a vector space under usual function addition and scalar multiplication.

A physics model allows displacement vectors \((x,y,z)\) only when \(z=5\). Explain algebraically why the allowed displacements do not form a vector space.

Let \(V\) be the set of all real \(2\times2\) matrices with trace \(0\). Show that \(V\) is a vector space under usual matrix operations.

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

For which real values of \(a\) is \(V_a=\{(x,y)\in\mathbb R^2:y=ax\}\) a vector space with usual operations?

For which real values of \(b\) is \(W_b=\{(x,y)\in\mathbb R^2:y=x+b\}\) a vector space with usual operations?

Let \(V_c=\{(x,y,z)\in\mathbb R^3:x+y+z=c\}\). Determine all \(c\in\mathbb R\) for which \(V_c\) is a vector space.

A student claims: "If a set contains \(\mathbf 0\), then it is a vector space." Disprove the claim with a concrete subset of \(\mathbb R^2\).

Suppose a non-empty subset \(V\subseteq\mathbb R^n\) is closed under addition and scalar multiplication. Prove that \(V\) contains the zero vector and additive inverses.

A set \(V\) uses usual vector addition but defines scalar multiplication by \(\lambda\odot\mathbf v=\mathbf v\) for every scalar \(\lambda\). Explain which vector-space axiom fails for a non-zero \(\mathbf v\).