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. Use this list to guide your study, not the sample exam mentioned below. 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).

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

Span{v₁, ..., v_p}

The definition of Ax in both words and symbols

Linearly independent set, linearly dependent set

Linear transformation

Standard matrix of a linear transformation

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The definition of Ax in both words and symbols

The definition of a matrix product AB

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)

 

Chapter 2:

Theorems 4, 5, 6, and 7 (first part)

Know the proof of Theorem 5

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 described by one or two geometric transformations.
• Use the definition of AB to deduce information about one matrix from another. (See Exercises 19-22 in Section 2.1.)
• Use an inverse matrix to solve a system of linear equations
• Find the inverse of a 2×2 matrix using Theorem 4 on p.110
• Find the inverse of a matrix A by row reducing [A   I].
• Use matrix algebra to solve equations involving matrices.
• Use the IMT to deduce some property of a square matrix when some other property of the matrix is given. (See Exercises 16-22 in Section 2.3.)

• 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.)
• Set up a migration matrix and write the difference equation x_k+1 = Mx_k that describes population movement in a region (assuming no births/deaths and no migration).
  Compute x₁ or x₂ when x₀ is specified, and interepret the results.