After completing this chapter, students should be able to do the following.
Objectives
Define a vector in \(\mathbb{R}^n \) as an ordered list of numbers, and understand the geometric interpretation of vectors in \(\mathbb{R}^2 \) and \(\mathbb{R}^3 \text{.}\)
Compute the magnitude of a vector and the unit vector in the direction of a given vector.
Perform vector operations in \(\mathbb{R}^n \text{,}\) including scalar multiplication, addition, and linear combinations.
Express vectors as linear combinations of other vectors in \(\mathbb{R}^n \text{.}\)
Describe the span of a set of vectors in \(\mathbb{R}^n \text{.}\)
Interpret a system of linear equations as a vector equation.
Define the matrix-vector product as a linear combination of the columns of the matrix.
Express a solution to a system of linear equations as a sum of a particular solution and a solution to the associated homogeneous system.
Determine linear independence / dependence of a set of vectors in \(\mathbb{R}^n \text{.}\)
Interpret a matrix as a linear transformation from \(\mathbb{R}^n\) to \(\mathbb{R}^m\text{.}\)
Produce a matrix representation of a linear transformation given its action on the standard unit vectors.
Identify some basic geometric transformations of the plane.
