MATH 461
The exam will cover the material we have discussed in class and studied in homework, from Sections 5.1- 5.7, Sections 6.1 - 6.6 and Sections 7.1 and 7.2.
The following list points out the most important definitions and theorems.
Note that when a computation is required, you may be asked to justify the conclusion you draw from the computation (by mentioning a fact or theorem, for example)
Definitions:
- Similar matrices, diagonalizable matrix
- Matrix of a linear transformation
- Inner product, length (or norm) of a vector, orthogonality
- Orthogonal complement
- Orthogonal sets, orthonormal sets
- Orthogonal basis, orthonormal basis
- Orthogonal projections
- Gram-Schmidt process
- QR factorization
- Least squares problem, normal equations
- Symmetric matrices
- Quadratic forms
- Classification of quadratic forms
Theorems:
Chapter 5:
- Theorem 2 (Linear independence of eigenvectors corresponding to distinct eigenvalues)
- Theorem 4 (Similar matrices have the same characteristic polynomial)
- Theorems 5 & 6 (Diagonalization)
- Theorem 8 (Diagonal matrix representation)
- Theorem 9 ('Hidden rotations')
Chapter 6:
- Theorem 2 (Pythagorean theorem)
- Theorem 4 (Linear independence of nonzero orthogonal vector)
- Theorems 5 (Orthogonal decomposition)
- Theorems 6,7 (Orthogonal matrices)
- Theorems 8,9,10 (Orthogonal projections)
- Theorem 11 (Gram-Schmidt)
- Theorem 13 (Normal equations)
Chapter 7:
- Theorem 1, 2
- Theorem 4, 5
Important Skills (partial list):
- Find the eigenvalues/eigenvectors of a matrix
- Diagonalize a matrix.
- Find the complex eigenvalues and corresponding eigenvectors of a 2× 2 matrix A.
- Find a factorization of a 2x2 matrix with a complex eigenvalue (section 5.5)
- Find the solution of a difference equation xk+1= A xk using the eigenvectors and eigenvalues of A. Be able to classify the origin as an attractor, a repellor or a saddle point.
- Compute the length of a vector. Normalize a vector.
- Check a set for orthogonality.
- Find the coordinates of a vector relative to an orthogonal basis.
- Compute the orthogonal projection of one vector onto another.
- Compute the orthogonal projection of a vector onto a subspace. Find the distance from a vector to a subspace. Decompose a vector as in the Orthogonal Decomposition Theorem.
- Apply the Gram-Schmidt process to a set of vectors.
- Solve a least squares problem by using the normal equations.
- Find the design matrix and observation vector for a least squares fitting problem (Section 6.6).
- Decide if a matrix is symmetric.
- Orthogonally diagonalize a symmetric matrix.
- Find the quadratic form associated to a symmetric matrix.
- Find the symmetric matrix associated to a quadratic form.
- Find the change of variable that transforms a given quadratic form into a quadratic form with no cross-product terms.
- Classify a quadratic form as positive definite, negative definite or indefinite.