# Review Questions for Vectors, matrices and norms

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