Linear Algebra

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Linear Algebra

• Chapter 2 Solving Linear Systems

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Chapter 2 Solving Linear Systems

• The primary goal is to deal with loose ends from
  the first set of lectures
• Develop systematic procedure for solving linear
  systems that works directly with the matrix form
• Establish conditions for a solution to a linear
  system to exist
• Establish conditions for a unique solution to a
  linear system
• Re-examine solutions to homogeneous systems
• Develop a systematic procedure for computing A1
• Discover equivalent conditions to nonsingularity

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Chapter 2 Solving Linear Systems

• 2.1 Echelon Form of a Matrix
• 2.2 Solving Linear Systems
• 2.3 Elementary Matrices Finding A-1
• 2.4 Equivalent Matrices
• 2.5 LU-Factorization

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2.1 Echelon Form of a Matrix

• Solve a system using elimination of variables

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2.1 Echelon Form of a Matrix

• Use back substitution to get
• x3 -2 gt x2 2 gt x1 1
• Solution (1, 2, -2)

Now use matrix form to solve system
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2.1 Echelon Form of a Matrix

• Use row operations to simplify using pivots and
  pivot rows and columns.

row echelon form (REF)
reduced row echelon form (RREF)
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2.1 Echelon Form of a Matrix

• Defn - An m x n matrix A is said to be in row
  echelon form (REF) if it has the following
  properties
• a) All zero rows, if any, appear at the bottom
  of the matrix.
• b) The first nonzero entry of a nonzero row is a
  1. This is
• called the leading one of its row.
• c) For each nonzero row, the leading one is to
  right and below all leading ones of preceding
  rows. (upside down staircase)
• Defn - If an m x n matrix A is to be in reduced
  row echelon form (RREF) if it satisfies
  properties (a), (b) (c), and
• d) If a column contains a leading one, then all
  the other entries in that column are zero.

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2.1 Echelon Form of a Matrix

• Examples - Row Echelon Form

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2.1 Echelon Form of a Matrix

• Examples - Reduced Row Echelon Form

Leading ones are the only nonzero entry in their
respective columns.
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2.1 Echelon Form of a Matrix

• Examples - Not Reduced Row Echelon Form

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2.1 Echelon Form of a Matrix

• Defn - An elementary row operation on a matrix A
  is any one of the following operations
• I) Interchange rows i and j of matrix A, Ri
  lt--gt Rj
• II) Multiply row i of A by c ? 0, cRi --gt Ri
• III) Add c times row i of A to row j of A, cRiRj
  --gt Rj

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2.1 Echelon Form of a Matrix

• Defn - An m x n matrix B is row equivalent to an
  m x n matrix A if B can be obtained by applying a
  finite sequence of elementary row operations to
  A. A B.
• This is an equivalence relation
• A A
• If A B, then B A.
• If A B and B C, then A C

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2.1 Echelon Form of a Matrix

• Theorem - Every nonzero m x n matrix A is row
  equivalent to a matrix in row echelon form.
• Reduction Process
• 1. Choose a pivot element from the nonzero
  entries in the 1st column. Row containing pivot
  call the pivot row.
• 2. Interchange rows (if nec) so that pivot row is
  the new 1st row.
• 3. Multiply pivot row by a constant s.t. the new
  pivot is 1.
• 4. Remove lower column entries i using the pivot
  row
• Ri ai,j R1-gt Ri
• 5. Repeat process with next column.

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2.1 Echelon Form of a Matrix

• Theorem - Every nonzero m x n matrix A is row
  equivalent to a unique matrix in reduced row
  echelon form.
• Reduction Process
• 1. Choose a pivot element from the nonzero
  entries in the 1st
• column. Row containing pivot call the pivot
  row.
• 2. Interchange rows (if nec) so that pivot row is
  the new 1st row.
• 3. Multiply pivot row by a constant s.t. the new
  pivot is 1.
• 4. Remove lower and upper column entries i using
  the pivot row
• Ri ai,j R1-gt Ri
• 5. Repeat process with next column.

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2.1 Echelon Form of a Matrix

• Can define column echelon form and reduced column
  echelon form in a similar manner. Basically, they
  are just the transpose of the corresponding row
  forms
• Put A in reduced column echelon form
• Work with AT put in RREF B
• BT is A in RCEF
• Close connection between echelon forms and the
  solution of linear equations