Review Questions for Vectors, matrices and norms

1. What is a vector space?
2. What is an inner product?
3. Given a specific function \(f(\mathbf{x})\), can \(f(\mathbf{x})\) be considered an inner product?
4. What is a vector norm? (What properties must hold for a function to be a vector norm?)
5. Given a specific function \(f(\mathbf{x})\), can \(f(\mathbf{x})\) be considered a norm?
6. What is the definition of an induced matrix norm? What do they measure?
7. What properties do induced matrix norms satisfy? Which ones are the submultiplicative properties? Be able to apply all of these properties.
8. For an induced matrix norm, given \(\|\mathbf{x}\|\) and \(\|{\bf A}\mathbf{x}\|\) for a few vectors, can you determine a lower bound on \(\|{\bf A}\|\)?
9. What is the Frobenius matrix norm?
10. For a given vector, compute the 1, 2 and \(\infty\) norm of the vector.
11. For a given matrix, compute the 1, 2 and \(\infty\) norm of the matrix.
12. Know what the norms of special matrices are (e.g., norm of diagonal matrix, orthogonal matrix, etc.)