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# 3: Linear Transformations and Matrix Algebra

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Learn about linear transformations and their relationship to matrices.

In practice, one is often lead to ask questions about the geometry of a transformation: a function that takes an input and produces an output. This kind of question can be answered by linear algebra if the transformation can be expressed by a matrix.

• 3.0: Prelude to Linear Transformations and Matrix Algebra
In this chapter, we will be concerned with the relationship between matrices and transformations.
• 3.1: Matrix Transformations
• 3.2: One-to-one and Onto Transformations
In this section, we discuss two of the most basic questions one can ask about a transformation: whether it is one-to-one and/or onto. For a matrix transformation, we translate these questions into the language of matrices.
• 3.3: Linear Transformations
In this section, we make a change in perspective. Suppose that we are given a transformation that we would like to study. If we can prove that our transformation is a matrix transformation, then we can use linear algebra to study it. This raises two important questions: (1) How can we tell if a transformation is a matrix transformation? (2) If our transformation is a matrix transformation, how do we find its matrix?
• 3.4: Matrix Multiplication
In this section, we study compositions of transformations. As we will see, composition is a way of chaining transformations together. The composition of matrix transformations corresponds to a notion of multiplying two matrices together. We also discuss addition and scalar multiplication of transformations and of matrices.
• 3.5: Matrix Inverses
In this section, we learn to “divide” by a matrix. This allows us to solve the matrix equation Ax=b in an elegant way.
• 3.6: The Invertible Matrix Theorem

3: Linear Transformations and Matrix Algebra is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Dan Margalit & Joseph Rabinoff via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.