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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.

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