The exam will cover the material we have discussed in class and studied in homework, from Chapter 1 and Sections 2.1 to 2.3. 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).

The definition of Ax in both words and symbols

Span{v}, Span {u,v} and geometric interpretation in R² or R³

Span{v₁, ..., v_p}

Linearly independent, linearly dependent

Linear transformation

Standard matrix of a linear transformation

The definition of a matrix product AB

The definition of the inverse of a matrix

Chapter 1:

Theorem 2 (Existence and Uniqueness Theorem)

Theorem 3 (Matrix equation, vector equation, system of linear equations)

Theorem 4 (When do the columns of A span Rⁿ ?)

Theorem 5 (Properties of the Matrix-Vector Product Ax)

Theorems 7, 8, 9 (Properties of linearly dependent sets)

Theorems 11 and 12 (one-to-one and onto linear transformations).

 

Chapter 2:

Theorems 4, 5, 6, and 7

Know the proof of Theorem 5

Theorem 8 (the Invertible Matrix Theorem)

• Know the difference between the concepts of matrix equation, vector equation, system of linear equations, and the augmented matrix for a system of linear equations. Given any one of these forms, be able to write one of the other equivalent forms.
• Determine when a system is consistent. Write the general solution in parametric vector form.
• Determine values of parameters that make a system consistent, or make the solution unique.
• Describe existence or uniqueness of solutions in terms of pivot positions.
• Determine when a homogeneous system has a nontrivial solution.
• Determine when a vector is in a subset spanned by specified vectors.
• Exhibit a vector as a linear combination of specified vectors.
• Write the equation of a line or plane in parametric vector form.
• Determine whether the columns of an m×n matrix span Rⁿ.
• Determine whether the columns are linearly independent.
• Determine whether a set of vectors is linearly independent. Know several methods that can sometimes produce an answer "by inspection" (without much calculation).
• Use linearity of matrix multiplication to compute A(u + v) or A(cu).
• Determine whether a specified vector is in the range of a linear transformation.
• Compute the image of a vector under a linear transformation.
• Find the standard matrix of a linear transformation.
• Determine whether a linear transformation is one-to-one or maps Rⁿ onto R^m
• Use an inverse matrix to solve a system of linear equations
• Find the inverse of a 2×2 matrix using Theorem 4 on p.119
• Find the inverse of a matrix A by row reducing [A   I].
• Use matrix algebra to solve equations involving matrices.

• Use linear combinations of vectors to describe various problems. (See Example 7 and Exercises 27, 28 in Section 1.3, Exercises 39, 40 in Section 1.5, and Example 6 in Section 1.7.)