The exam will cover the material we have discussed in class and studied in homework, from Chapter 3, Sections 4.1 to 4.7, and sections 5.1, 5.2. 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:

• Vector space, subspace, null space, column space
• Linear transformation
• Linear independence, linear dependence, linear dependence relation (for vector spaces)
• Spanning set, basis, coordinate vector
• Dimension of a vector space
• Rank of a matrix
• Change of coordinates matrix
• Eigenvalue, eigenvector, eigenspace
• Characteristic polynomial, characteristic equation
• Similar matrices

• Theorems 2, 3, 4, 5, 6 (properties of determinants)
• The formula for det A as a product of pivots (page 188)
• Theorem 7 (Cramer's Rule)
• Theorems 9 and 10 (determinants as area or volume)

• Theorems 1-4, 5 (Spanning Set Theorem), 6, 7 (Unique Representation Theorem), 8-10, 12 (Basis Theorem), 13, 14 (Rank Theorem) and 15.
• Know the proofs of Theorems 2 and 7.
• The Invertible Matrix Theorem (including new statements in Sections 3.2 (Theorem 4) and 4.6)

• Theorem 2 (Linear independence of eigenvectors corresponding to distinct eigenvalues)
• Theorem 4 (Similar matrices have the same characteristic polynomial)

• Compute a determinant using a combination of elementary row (or column) operations and cofactor expansions (across a row or down a column).
• Compute a 3 by 3 determinant.
• Compute areas or volumes of simple objects using determinants.
• Determine if a set of vectors spans (or is a basis for) Rⁿ.
• Determine if a set is a subspace (using Theorems 1, 2, or 3 in Chapter 4).
• Determine if a vector is in Nul A or in Col A.
• Find a nonzero vector in Nul A or Col A.
• Determine if a set is a basis for a subspace.
• Find a basis for Nul A, Col A, or other subspaces.
• Find the coordinate vector of a vector relative to a basis.
• Find the change-of-coordinates matrix.
• Use coordinate vectors to check if a set is linearly independent.
• Find the dimension of Nul A, Col A, Row A, or other subspace.
• Find a basis for Row A.
• Find a basis for Nul A, Col A, Row A, or other subspaces.
• Determine the rank of a matrix.
• Use the Rank Theorem to determine facts about a system of linear equations.
• Find a change of coordinate matrix
• 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.