# mat 2401 linear algebra

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MAT 2401 Linear Algebra. 2.1 Operations with Matrices. http://myhome.spu.edu/lauw. Today. WebAssign 2.1 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. - PowerPoint PPT PresentationTRANSCRIPT

MAT 2401Linear Algebra2.1 Operations with Matriceshttp://myhome.spu.edu/lauw

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.

TodayWritten HWAgain, today may be longer. It is more efficient to bundle together some materials from 2.2. Next class session will be shorter.

PreviewLook at the algebraic operations of matrices term-by-term operationsMatrix Addition and SubtractionScalar Multiplication Non-term-by-term operationsMatrix Multiplication

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

NotationsMatrix

NotationsMatrix

Example:

Special CasesRow Vector

Column Vector

Matrix Addition and SubtractionLet A = [aij] and B = [bij] be mxn matrices

Sum: A + B = [aij+bij]Difference: A-B = [aij-bij]

(Term-by term operations)

Example 1

Scalar MultiplicationLet A = [aij] be a mxn matrix and c a scalar.

Scalar Product: cA=[caij]

Example 2

Matrix MultiplicationDefine multiplications between 2 matricesNot term-by-term operations

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

MotivationSince 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 FactsThe 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 Matrixnxn Square Matrix

Zero Matrixmxn Matrix with all zero entries

Representation of Linear System by Matrix Multiplication

Representation of Linear System by Matrix Multiplication

Representation of Linear System by Matrix Multiplication

Representation of Linear System by Matrix Multiplication

Let

Then the linear system is given by

RemarkIt would be nice if division can be defined such that:

(2.3) Inverse

Let

Then the linear system is given by

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.