The exam will cover the material we have discussed in class and studied in homework, from Sections 5.1-5.6 and Sections 6.1 to 6.6. 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)

• Eigenvalue, eigenvector, eigenspace
• Characteristic polynomial, characteristic equation
• 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
• The QR factorization will not be on the test
• Least squares problem, normal equations

• 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')

• 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)

• Decide whether a vector is an eigenvector of a matrix.
• Find the eigenvalues of a triangular matrix
• Find the eigenvectors corresponding to a known eigenvalue. Find a basis for an eigenspace.
• Find the eigenvalues and eigenvectors of a 2× 2 matrix A.
• Diagonalize a matrix given its eigenvalues.
• Find the matrix of a linear transformation (e.g p.333:1,p.334:11).
• Find the complex eigenvalues and corresponding eigenvectors of a 2× 2 matrix A.
• Find a factorization of a 2x2 matrix with a complex eigenvalue (e.g. p. 341:13)
• Find the solution of a difference equation x_k+1= A x_k 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 subsapce. 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.