Stanford ENGR108: Introduction to Applied Linear Algebra | 2020 | Lecture 28-VMLS 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

we're now going to talk about composition of linear functions we'll see that that gives yet another interpretation of matrix multiplication so let's suppose a is an m by p matrix and b is p by n so these are two matrices um of course i can multiply i can form the product c equals a b so let's define a function f and it's going to map rp to rm and t... Read More

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.