Do we really need four types of matrix-vector multiplication?
We explore the algebraic properties of row-vector-matrix multiplication.
00:00 -Introduction
00:05 -Matrix-vector multiplication is actually two operations
00:12 -Operation 1: Matrix-column-vector multiplication
00:43 -Operation 2: Row-vector-matrix multiplication
01:12 -The inner dimensions must agree
01:26 -We can form the each output in two different ways
01:33 -Vectorize the data : generate output as a linear combo of vectors
01:40 -Scalarize the data : generate output as a set of individual entries
01:48 -Transpose operator shows how four versions are interrelated
02:08 -Transpose Theorem: Transpose of a Matrix-Vector Product
03:07 -VANVISA model for deep learning in mathematics
04:24 -A theorem without a proof is like a joke without a punchline
04:31 -To strengthen your math genius, prove every theorem for yourself
04:54 -When analyzing theorem, be mindful of conditional statements
05:16 -For any conjecture, we can either prove or provide a counter example
05:23 -How to Read and Do Proofs by Daniel Solow
05:33 -Different approaches to prove a conditional statement
06:10 -What is a counter example?
06:34 -Let’s use a direct proof for the transpose theorem
06:44 -Identify all conditional statements, antecedents, and consequents
06:57 -Proposition 1: Transpose of a matrix-column-vector product
07:16 -Proposition 2: Transpose of a row-vector-matrix product
07:48 -Deep learning habit: state math in intuitive and nerdy language
08:07 -Let’s do two different direct proofs of proposition 1
08:18 -Return to our key questions from problem solving
https://jeffandersonmath.wordpress.co...
08:34 -Create a simpler problem first as we work toward general proof
08:42 -Recall the forward approach and the backward approach to proofs
09:07 -Create a simpler example to get insights into the general case
“Hard enough to be interesting but easy enough to be quickly”
09:47 -The forward approach to Proposition 1 via dot products
10:53 -Active recall to strengthen your brain
11:33 -Definition of matrix-column-vector mult via dot products
12:44 -The formula for the kth entry of matrix-column-vector product
13:23 -The backward approach to Proposition 1 via dot products
14:30 -The formula for kth entry of row-vector-matrix product
14:36 -Unite the forward and backward approaches
15:19 -Generating the intuition behind proposition 1
15:53 -The general proof of Proposition 1 via dot products
17:51 -Deep learning habit: look for multiple proofs for same result
18:30 -Develop a vectorized proof of Proposition 1
19:11 -Analyze the simpler example to get insights into the general case
19:39 - Definition of matrix-column-vector mult via linear combo
19:50 -Apply definition for simple example
20:26 -Recall the algebraic properties of the vector transpose
21:06 -Finish example proof of Proposition 1 via linear combo
21:54 -General approach to generate formal proof of Proposition 1
23:11 -Challenge 1: Generate a third proof of Proposition 1
23:19 -Challenge 2: Generate more than one proof of Proposition 2
24:02 -What does all this mean for create computer code?
25:25 -Conclusion