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Applied Linear Algebra

Applied Linear Algebra. Review. LU Factorization. If a square matrix can strict upper triangular form, U, without interchanging any rows, then A can be factored as A=LU, where L is a low triangular matrix. Ax= b L(Ux )= b Solve Ly= b then y = Ux. Elementary Matrices.

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Applied Linear Algebra

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  1. Applied Linear Algebra Review

  2. LU Factorization • If a square matrix can strict upper triangular form, U, without interchanging any rows, then A can be factored as A=LU, where L is a low triangular matrix. Ax=b L(Ux)=b Solve Ly=b then y=Ux

  3. Elementary Matrices • An elementary matrix of type I is a matrix obtained by interchanging two rows of I

  4. Elementary Matrices Row 1 and Row 2 are switched! Column 1 and Column 2 are switched!

  5. Elementary Matrices • An elementary matrix of type I is a matrix obtained by interchanging two rows of I • An elementary matrix of type II is a matrix obtained by multiplying a row of I by a nonzero constant

  6. Elementary Matrices Row 2 is multiplied by 3! Column 2 is multiplied by 3!

  7. Elementary Matrices • An elementary matrix of type I is a matrix obtained by interchanging two rows of I • An elementary matrix of type II is a matrix obtained by multiplying a row of I by a nonzero constant. • An elementary matrix of type III is a matrix obtained from I by adding a multiple of one row to another row.

  8. Elementary Matrices 4 times Row 3 is added to Row 1 4 times Column 1 is added to column 3

  9. Change of Basis

  10. Change of Basis

  11. Gram-Schmidt Orthonomalization Process

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