The Exam will cover the material we have discussed in class and studied in homework from Chapter 3 and Chapter 4 (omitting Section 4.8). The following list points out the main definitions and theorems.

• Subspace, null space, column space
• Linear transformation
• Linear independence, linear dependence, linear dependence relation (for vector spaces)
• Basis, coordinate vector
• Dimension of a vector space
• Rank of a matrix
• Change of coordinates matrix
• Probability vector (state vector), stochastic matrix, steady-state vector

• 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)

• Compute a determinant using a combination of elementary row (or column) operations and cofactor expansions (across a row or down a column).
• 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.

• Compute areas or volumes of simple objects using determinants.
• Set up a stochastic matrix for an applied problem. Determine the steady state vector and use it to describe the long-term behavior of the system modeled by a Markov chain.