How to do row vector matrix multiplication via linear combinations?
We investigate the formal definition of row-vector-matrix multiplication via linear combinations:
00:00 -Introduction
00:05 -Let A be an m-by-n matrix
00:13 -Let x be an m-by-1 column vector
00:18 -The number of rows of x matches the number of rows of A
00:23 -The linear combination version of row-vector-matrix multiplication
00:41 -Important question: why do we assume x is a column vector?
01:00 -Scale each row of matrix A by the correspond entry of vector x
01:46 -How to use colon notation to define row-vector-matrix multiplication
02:09 -Useful habit: Write the dimensions as subscripts to check dimensions
02:47 -When the modeling matrix shows up on the right, cut that into rows
03:13 -Why must the inner dimensions agree?
03:32 -Why is the size of the output determined by the size of each row?
04:03 -Vectorize our data: Chunk data into vector-sized pieces
04:57 -Scalarize our data: look at each individual entry
06:21 -Using linear combinations to generate a formula for the individual entries
07:09 -Community Challenge 1: Come up with the dot product version
07:40 -Two special features of this definition
07:46 -Feature 1: Connect matrix partitions to location of modeling matrix
08:59 -When modeling matrix is the right factor, chop it into rows
09:24 -Major theme: when do we use column or row partitions?
09:41 -Why do we use a transpose to define row-vector-matrix multiplication?
11:57 -Conclusion