Content Goals – for Chapter 1
Content Goals – for Chapter 2
Sections: 2.0, 2.1, 2.2 (thru p. 80, skipping Linear Systems over Zp), 2.3
• Recognize linear versus nonlinear equations
• Solve systems of linear equations with constant coefficients using
o A graphical approach
▪ As intersection of lines and planes
▪ As linear combination of vectors [note that this is what Theorem 2.4 in section 2.3 is stating; and is what Problems 3 & 4 in section 2.0 were highlighting]
o A set of algebraic equations
o The corresponding augmented matrix, and
▪ Gaussian elimination (and back substitution)
▪ Gauss-Jordan elimination (and back substitution)
• Understand that these systems have 0, 1, or infinite number of solutions
• Elementary row operations on a system do not change the solutions to the system
• Row echelon form and reduced row echelon form of a matrix
• Reduced row echelon form is unique
• Rank of a matrix, and the Rank Theorem
• Homogeneous systems must have 1 or an infinite number of solutions (never 0 solutions), and the Rank Theorem tells us which
• Definition of linear dependence and linear independence
o Note that Theorems 2.5-2.7 give equivalent ways of stating a set of vectors are linearly dependent, but they also connect important concepts in this section to the idea of linear dependence/independence
o Know how to determine if a set of vectors are linearly dependent or independent, and if the former, an expression for the dependence
• Span of a set of vectors and the spanning set of a space
o Note that later in the book (section 3.5) we’ll see that a “basis” is a slightly more restrictive set than the spanning set; but it’s really the set we care about
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