1. The statement says: If $\{ \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \}$ is a basis for a vector space $V$, then for every vector $\mathbf{v}$ in $V$ there exists a unique linear combination of $\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n$ equal to $\mathbf{v}$. This is true by definition of a basis: a basis is a linearly independent set that spans $V$, so every vector in $V$ can be uniquely expressed as a linear combination of basis vectors. Answer: T 2. The statement says: If $\{A_1, A_2, \ldots, A_{16}\}$ is a linearly independent set of $4 \times 4$ matrices, then it is a basis for $M_4(\mathbb{R})$. The dimension of $M_4(\mathbb{R})$ is $4 \times 4 = 16$, so any linearly independent set of 16 matrices forms a basis. Answer: T 3. The statement says: The vector space $C^{\infty}(-\infty, \infty)$ of all functions defined and continuous on $\mathbb{R}$ cannot be spanned by any finite set. Since $C^{\infty}(-\infty, \infty)$ is infinite-dimensional, no finite set can span it. Answer: T 4. The statement says: Every basis of $P_4(\mathbb{R})$, the space of polynomials degree $\leq 4$, must contain at least one polynomial of degree 3 or less. A basis for $P_4(\mathbb{R})$ can be $\{1, x, x^2, x^3, x^4\}$ which contains polynomials of degree 0,1,2,3,4. So it must contain polynomials of degree 3 or less. Answer: T 5. The statement says: If $\mathbf{x} \in \mathbb{R}^n$ and $\mathcal{B}$ is any basis for $\mathbb{R}^n$, then the coordinate vector of $\mathbf{x}$ with respect to $\mathcal{B}$ is the same as $\mathbf{x}$. This is false because the coordinate vector depends on the basis $\mathcal{B}$ and generally differs from $\mathbf{x}$ unless $\mathcal{B}$ is the standard basis. Answer: F