1. **Problem:** Determine the truth value of each statement and justify with proof or counter-example. 2. **(a) Eigenvalues of a symmetric matrix are real.** - **Fact:** A symmetric matrix $A$ satisfies $A = A^T$. - **Theorem:** All eigenvalues of a real symmetric matrix are real. - **Reason:** Symmetric matrices are diagonalizable by an orthogonal matrix, and their eigenvalues are real. - **Conclusion:** True. 3. **(b) If $S_1 \subset S_2$ are subsets of a vector space $V$ and $S_2$ is linearly independent, then $S_1$ is also linearly independent.** - **Reason:** Any subset of a linearly independent set is also linearly independent. - **Conclusion:** True. 4. **(c) For any linear transformation $T: \mathbb{R}^4 \to \mathbb{R}^5$, $\ker(T) \neq \{0\}$.** - **Reason:** By the Rank-Nullity Theorem, $\dim(\ker(T)) = 4 - \operatorname{rank}(T)$. - Since $\operatorname{rank}(T) \leq 4$, the kernel can be zero only if $T$ is injective. - But $T$ maps from a 4-dimensional space to a 5-dimensional space, so injective maps exist. - **Example:** The inclusion map $T(x) = (x,0)$ is injective with kernel zero. - **Conclusion:** False. 5. **(d) Every unitary matrix is Hermitian.** - **Definition:** A unitary matrix $U$ satisfies $U^* U = I$. - A Hermitian matrix $H$ satisfies $H = H^*$. - **Counter-example:** The unitary matrix $U = \begin{bmatrix}0 & 1 \\ -1 & 0\end{bmatrix}$ is not Hermitian. - **Conclusion:** False. 6. **(e) There exists a linear operator $T: \mathbb{R}^4 \to \mathbb{R}^4$ with characteristic polynomial $(x-3)^2 (x-5)(x-1)$ and minimal polynomial $(x-3)(x-5)$.** - **Fact:** The minimal polynomial divides the characteristic polynomial and contains all eigenvalues with their largest Jordan block size. - Here, minimal polynomial lacks $(x-1)$ factor, which is an eigenvalue from characteristic polynomial. - This is impossible because minimal polynomial must include all eigenvalues. - **Conclusion:** False. **Final answers:** (a) True (b) True (c) False (d) False (e) False