The Exam will cover the material we have discussed in class and studied in homework from Chapter 5, sections 1-5 & 7 and Chapter 6, sections 1-6. The following list points out the main definitions and theorems.

• Eigenvalue, eigenvector
• Characteristic polynomial, characteristic equation
• Similar matrices, diagonalizable matrix
• Matrix of a linear transformation
• Inner product, length, orthogonality
• Orthogonal compliment
• Orthogonal sets, orthonormal sets
• Orthogonal projections
• Gram-Schmidt process
• Least squares problem, normal equations

• Theorem 2 (Linear independence of orthogonal vectors)
• 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 3 (Orthogonal compliments)
• Theorems 5,6,7 (Orthogonal and orthonormal sets)
• 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 eigenvectors corresponding to a known eigenvalue.
• Find the eigenvalues and eigenvectors of a 2× 2 matrix A.
• Find the orthogonal compliment of a subspace of Rⁿ.
• Diagonalize a matrix given its eigenvalues.
• Find the matrix of a linear transformation (e.g p.333:1,p.334:11).
• Compute the orthogonal projection of one vector on another.
• Compute the orthogonal projection of a vector onto a subspace.
• Apply the Gram-Schmidt process to a set of vectors.
• Solve a least squares problem by using the normal equations.

• Solve an initial value problem for a 2×2 system of differential equations.
• Find the equation of the least-squares line that best fits a given set of data points.