1. Problem i: Find $r$ and $s$ such that $AB^T = 0$, where $A = [1, r, 1]$, $B = [-2, 2, s]$ Calculate $$AB^T = 1 \times (-2) + r \times 2 + 1 \times s = -2 + 2r + s = 0.$$ Rearranging, $$s = 2 - 2r.$$ Thus, $r$ and $s$ satisfy $s = 2 - 2r$. 2. Problem ii: Define symmetric matrix. A symmetric matrix is a square matrix $A$ that equals its transpose: $$A = A^T.$$ This means element $a_{ij} = a_{ji}$ for all $i$, $j$. 3. Problem iii: Prove uniqueness of inverse of a matrix if exists. Suppose $A$ has two inverses $B$ and $C$ such that $$AB = BA = I$$ and $$AC = CA = I,$$ where $I$ is the identity matrix. Then $$B = BI = B(AC) = (BA)C = IC = C,$$ showing the inverse is unique. 4. Problem iv: Find $(A^T)^{-1}$ for $$A = \begin{bmatrix}1 & 2 \\ 3 & 4\end{bmatrix}.$$ First, transpose: $$A^T = \begin{bmatrix}1 & 3 \\ 2 & 4\end{bmatrix}.$$ Find inverse: Determinant: $$det(A^T) = 1 \times 4 - 3 \times 2 = 4 - 6 = -2.$$ So, $$(A^T)^{-1} = \frac{1}{-2} \begin{bmatrix}4 & -3 \\ -2 & 1\end{bmatrix} = \begin{bmatrix}-2 & 1.5 \\ 1 & -0.5\end{bmatrix}.$$ 5. Problem v: Find unit vector in direction of $v = (3,4)$. Magnitude of $v$: $$||v|| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5.$$ Unit vector: $$\frac{1}{5}(3,4) = \left(\frac{3}{5}, \frac{4}{5}\right).$$ 6. Problem vi: Define orthogonal subspace. An orthogonal subspace to a subspace $W$ of a vector space $V$ consists of all vectors in $V$ that are orthogonal to every vector in $W$. 7. Problem vii: Find $A^H$ for $$A = \begin{bmatrix}1 & i \\ 0 & 1+i\end{bmatrix}.$$ $A^H$ is the conjugate transpose: Conjugate: replace $i$ by $-i$ $$A^* = \begin{bmatrix}1 & -i \\ 0 & 1 - i\end{bmatrix}$$ Transpose: $$A^H = \begin{bmatrix}1 & 0 \\ -i & 1 - i\end{bmatrix}.$$ 8. Problem viii: Define diagonal matrix. A diagonal matrix is a square matrix where all entries outside the main diagonal are zero. 9. Problem ix: For $u = (1, -2, c)$ and $v = (2, 1, -1)$ orthogonal means their dot product equals zero: $$1 \times 2 + (-2) \times 1 + c \times (-1) = 2 - 2 - c = 0$$ This implies $$-c = 0 \Rightarrow c = 0.$$ 10. Problem x: Define similar matrix. Two square matrices $A$ and $B$ are similar if there exists an invertible matrix $P$ such that $$B = P^{-1}AP.$$ They represent the same linear transformation under different bases. 11. Problem xi: Find $c$, $d$ such that $$c \begin{bmatrix}1 \\ 2 \end{bmatrix} + d \begin{bmatrix}3 \\ 1\end{bmatrix} = \begin{bmatrix}14 \\ 8 \end{bmatrix}.$$ Equations: $$1c + 3d = 14,$$ $$2c + 1d = 8.$$ From second: $$d = 8 - 2c.$$ Substitute in first: $$c + 3(8 - 2c) = 14 \Rightarrow c + 24 - 6c = 14 \Rightarrow -5c = -10 \Rightarrow c = 2.$$ Then $$d = 8 - 2 \times 2 = 4.$$ 12. Problem xii: Given invertible $A$ and $AB = AC$, prove $B = C$. Multiply both sides by $A^{-1}$: $$A^{-1}AB = A^{-1}AC \Rightarrow IB = IC \Rightarrow B = C.$$