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 Presentation

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• 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.