Select the true statements about zero vectors. There may be more than one correct answer.
A.
The zero vector in \( \mathcal{P}_2 \) is \( \langle 0,0,0 \rangle \).
B.
The zero vector in \( \mathcal{F}(\mathbb{R},\mathbb{R}) \) is \( f(0) = 0 \).
C.
The zero vector in \( \mathcal{P}_5 \) is \( f(t) = 0 \) for all real numbers \( t \).
D.
The zero vector in \( \mathcal{F}( \lbrace 1, 2, 3, 4 \rbrace, \mathbb{R} ) \) is the function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(1) = 0 \), \( f(2) = 0 \), \( f(3) = 0 \), and \( f(4) = 0 \).
E.
The zero vector in \( M_{2,2}(\mathbb{R}) \) is \( \left( \begin{array}{rr} 0 & 0 \\ 0 & 0 \end{array} \right) \).
F.
The zero vector in \( M_{2,2}(\mathbb{R}) \) is \( \langle 0, 0, 0, 0 \rangle \).
Select the true statements about vectors in vector spaces. There may be more than one correct answer.
A.
The sum \( \langle 2,3 \rangle + 4 \mathbf{e_1} \) is a vector in \( \mathbb{R}^2 \).
B.
The function \( f(t) = e^t \) is a vector in \( \mathcal{P}_{\infty} \).
C.
The additive inverse of the vector \( f(t) = 4 + 5t + 6t^2 \) in \( \mathcal{P}_2 \) is \( f(-t) \).
D.
The function \( f(t) = \ln(t) \) is a vector in \( \mathcal{F}(\mathbb{R},\mathbb{R}) \).
E.
If \( f(t) \) is in \( \mathcal{P}_2 \), then \( (f(t))^2 \) is in \( \mathcal{P}_2 \).
F.
The function \( f(t) = 2 + 3t \) is a vector in \( \mathcal{P}_5 \).
G.
If \( f(t) \) is in \( \mathcal{P}_2 \), then \( f(t^3) \) is in \( \mathcal{P}_2 \).
You can earn partial credit on this problem.