MAT 2401 Linear Algebra 2.1 Operations with Matrices

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Presentation transcript:

MAT 2401 Linear Algebra 2.1 Operations with Matrices

HW... If you do not get 9 points or above on #1, you are not doing the GJE correctly. Some of you are doing RE. GJE is the corner stone of this class, you really need to figure it out.

Today Written HW Again, today may be longer. It is more efficient to bundle together some materials from 2.2. Next class session will be shorter.

Preview Look at the algebraic operations of matrices “term-by-term” operations Matrix Addition and Subtraction Scalar Multiplication Non-“term-by-term” operations Matrix Multiplication

Matrix If a matrix has m rows and n columns, then the size (dimension) of the matrix is said to be mxn.

Notations Matrix

Notations Matrix Example:

Special Cases Row Vector Column Vector

Matrix Addition and Subtraction Let A = [a ij ] and B = [b ij ] be mxn matrices Sum: A + B = [a ij +b ij ] Difference: A-B = [a ij -b ij ] (Term-by term operations)

Example 1

Scalar Multiplication Let A = [a ij ] be a mxn matrix and c a scalar. Scalar Product: cA=[ca ij ]

Example 2

Matrix Multiplication Define multiplications between 2 matrices Not “term-by-term” operations

Motivation The LHS of the linear equation consists of two pieces of information: coefficients: 2, -3, and 4 variables: x, y, and z

Motivation Since both the coefficients and variables can be represented by vectors with the same “length”, it make sense to consider the LHS as a “product” of the corresponding vectors.

Row-Column Product

Example 3

Matrix Multiplication

Example 4

Example 5 (a) Scratch: Q: Is it possible to multiply the 2 matrices? Q: What is the dimension of the resulting matrix?

Example 5 (b) Scratch: Q: Is it possible to multiply the 2 matrices? Q: What is the dimension of the resulting matrix?

Example 5 (c) Scratch: Q: Is it possible to multiply the 2 matrices? Q: What is the dimension of the resulting matrix?

Example 5 (d) Scratch: Q: Is it possible to multiply the 2 matrices? Q: What is the dimension of the resulting matrix? Remark:

Example 5 (e) Scratch: Q: Is it possible to multiply the 2 matrices? Q: What is the dimension of the resulting matrix? Remark:

Example 5 (f) Scratch: Q: Is it possible to multiply the 2 matrices? Q: What is the dimension of the resulting matrix? Remark:

Interesting Facts The product of mxp and pxn matrices is a mxn matrix. In general, AB and BA are not the same even if both products are defined. AB=0 does not necessary imply A=0 or B=0. Square matrix with 1 in the diagonal and 0 elsewhere behaves like multiplicative identity.

Identity Matrix nxn Square Matrix

Zero Matrix mxn Matrix with all zero entries

Representation of Linear System by Matrix Multiplication

Let Then the linear system is given by Representation of Linear System by Matrix Multiplication

Let Then the linear system is given by Remark It would be nice if “division” can be defined such that: (2.3) Inverse

HW... If you do not get 9 points or above on #1, you are not doing the GJE correctly. Some of you are doing RE. GJE is the corner stone of this class, you really need to figure it out.