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Chapter 4 Matrices (MX)
In
Chapter 3 , we saw that matrices are fundamentally just a representation of linear transformations (with respect to a choice of basis). Here, we explore the algebraic structure of the set of matrices of a given size, which will provide us further tools for understanding linear transformations.
Motivating Question. What algebraic structure do matrices have?
Learning Outcomes By the end of this chapter, you should be able to...
Determine if a matrix is invertible, and if so, compute its inverse and use it to solve an appropriate system of equations.
Calculate the change-of-basis matrix for the standard basis to a non-standard basis of
\(\mathbb R^n\text{.}\)
Express row operations through matrix multiplication.
Readiness Assurance.
Before beginning this chapter, you should be able to...
Compose functions of real numbers.
Identify the domain and codomain of linear transformations.
Calculate the standard matrix of a linear transformation and use it to compute the image of a given vector under the transformation.
Determine if a linear transformation is injective and/or surjective.
Interpret the ideas of injectivity and surjectivity in multiple ways.
