Stanford ENGR108: Introduction to Applied Linear Algebra  2020  Lecture 28VMLS matrix mult ex  Summary and Q&A
TL;DR
Matrix multiplication is equivalent to composing linear functions, where the result is a linear function whose associated matrix is the product of the associated matrices of the input functions.
Key Insights
 ✖️ Matrix multiplication can be interpreted as the composition of linear functions.
 ❓ Composing two linear functions results in another linear function.
 🔠 The associated matrix of the composed function is the product of the associated matrices of the input functions.
 The order of matrices in matrix multiplication is significant.
 ❓ Linear functions can be represented by matrices to simplify calculations.
 ❓ The composition of linear functions is associative.
 ✖️ Matrix multiplication can be used to find second differences in a sequence of numbers.
Transcript
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Questions & Answers
Q: How can linear functions be represented using matrices?
Linear functions can be represented by matrices, where the input vector is multiplied by a matrix to produce the output vector.
Q: How is matrix multiplication related to the composition of linear functions?
Matrix multiplication can be interpreted as the composition of linear functions, where the output vector of one function is used as the input for another function.
Q: What is the result of composing two linear functions?
The composition of two linear functions results in another linear function, with the associated matrix being the product of the associated matrices of the input functions.
Q: Why does the order of matrices matter in matrix multiplication?
The order of matrices matters in matrix multiplication because the first matrix in the product operation is the one that directly affects the input vector.
Summary & Key Takeaways

Linear functions can be represented by matrices.

Matrix multiplication can be interpreted as the composition of linear functions.

The composition of two linear functions results in another linear function, with the associated matrix being the product of the input function matrices.