Day 2

Summary of Day 1

Nothing can stop the man with the right mental attitude from achieving his goal; nothing on earth can help the man with the wrong mental attitude.
Thomas Jefferson

Basic Definitions 

 

An m×n matrix A is a rectangular array of real numbers with m rows 

and n columns. (Rows are horizontal and columns are vertical.) The

numbers m and n are the dimensions of A.The real numbers in the 

matrix are called its entries. The entry in row i and column j is 

called a_ij or A_ij.

Example

Following is a 4×5 matrix with the entry A₂₃ highlighted. 

A =		0	1	2	0	3
		1/3	-1	10	1/3	2
		3	1	0	1	-3
		2	1	0	0	1

Operations with Matrices

Transpose

The transpose, A^T, of a matrix A is the matrix obtained from A

by writing its rows as columns. If A is an m×n matrix and B = A^T, 

then B is the n×m matrix with b_ij = a_ji. 

Examples  

Transpose

0 1 2 T 1/3 -1 10

=

0 1/3 1 -1 2 10

Sum, Difference If A and B have the same dimensions, then their sum, A+B, is obtained by adding corresponding entries. In symbols, (A+B)ij = Aij + Bij. If A and B have the same dimensions, then their difference, A - B, is obtained by subtracting corresponding entries. In symbols, (A-B)ij = Aij - Bij. Scalar Multiple If A is a matrix and c is a number (sometimes called a scalar in this context), then the scalar multiple, cA, is obtained by multiplying every entry in A by c. In symbols, (cA)ij = c(Aij). Sum & Scalar Multiple 0 1 1/3 -1 + 2 1 -1 2/3 -2 = 2 -1 5/3 -5 Product If A has dimensions m×n and B has dimensions n×p, then the product AB is defined, and has dimensions m×p. The entry (AB)ij is obtained by multiplying row i of A by column j of B, which is done by multiplying corresponding entries together and then adding the results.  Product 0 1 1/3 -1 1 -1 2/3 -2 = 2/3 -2 -1/3 5/3