📘 Linear Algebra

1. **Find the angle between vectors** $\mathbf{u} = \begin{bmatrix}3 \\ 2 \\ 1\end{bmatrix}$ and $\mathbf{v} = \begin{bmatrix}1 \\ 0 \\ -1\end{bmatrix}$ using the dot product formu

1. Stating the problem: Consider matrix $A = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$. We need to find determinants for $A$, $4A$, and $\begin{bmatrix} 11 & 1 \\ -9 & -1 \end

1. **QUESTION ONE** **a) Problem:** Determine values of $x$ such that matrix

1. **Problem Statement:** Given matrices:

1. **Problem Statement:** Find the eigenvalues and corresponding eigenvectors of the matrix: $$A = \begin{bmatrix} 1 & 1 & 3 \\ 1 & 5 & 1 \\ 3 & 1 & 1 \end{bmatrix}$$

1. **Problem Statement:** Find the determinant of the matrix $$A = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}$$. 2. **Recall the determinant formula for a 2x2 matrix:** For $$A

1. **Problem Statement:** Find matrix $A$ for the quadratic form $2xy - 2yz + 2zx$. Then find eigenvalues and normalized eigenvectors. Finally, diagonalize matrix $A$, convert it t