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| − | == Lecture 4 (01/21/10) == | + | == Lecture 4 (01/21/10) == |
| − | == Row Echelon Form (ref) == | + | == Row Echelon Form (ref) == |
| − | + | ||
| − | + | [[Definition:]]Let A be a matrix, A will be a row echelon form(ref) if: | |
| − | + | 1. If any, a row full of zeros has to be at the bottom. | |
| − | + | 2. The leftmost nonzero in any row is "1", also known as 'leading 1's' | |
| − | + | 3. If row i and j are nonzero and i<j, the the 'leading 1' in row i is to the left of'leading 1'in row j | |
| − | + | [[Note:]]Numbers following the 'leading 1's' can be any numbers | |
| − | [[ | + | |
| − | + | == Reduced Row Echelon Form (rref) == | |
| − | 2. Any number above 'leading's 1' can only be zeros | + | [[Definition:]]Matrix A is in reduced row echelon form(rref) if: |
| + | |||
| + | 1. A is in row echelon form(ref) | ||
| + | |||
| + | 2. Any number above 'leading's 1' can only be zeros | ||
[[Note:]]Every number in the column above the 'leading 1' need to be zeros | [[Note:]]Every number in the column above the 'leading 1' need to be zeros | ||
| − | == Elementary Transformation Steps for rref Conversion == | + | == Elementary Transformation Steps for rref Conversion == |
| + | |||
| + | 1. Switching rows | ||
| + | |||
| + | 2. Scale rows with any number | ||
| + | |||
| + | 3.Take any row and add a scale version of any other row to it. | ||
| + | |||
| + | [[Note:]]It is always a good idea to try to rearrange rows to make the matrix easier to convert first and tries to avoid creating any fractions, especially in the early stages. | ||
| + | |||
| + | <br> | ||
| + | |||
| + | <br> Properties of a Determinant | ||
| + | |||
| + | 1. det(A) = det(transpose of A) | ||
| + | |||
| + | 2. det(A with rows i and j interchanged) = -det(A) | ||
| − | + | 3. det(A with row j replaced by row j +c*rowi) = det(A) | |
| − | + | 4. det(matrix with a row of zeros) = 0 | |
| − | + | 5. det(matrix with 2 equal rows) = 0 | |
| − | + | - All of these properties stay true if you replace row with column | |
| + | 6. det(AB) = det(A)*det(B) | ||
| + | 7. if det(A) = 0 then it has no inverse | ||
| − | + | 8. det(BA(inverseB)) = det(A) | |
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | ||
| − | + | ||
| − | det(BA(inverseB)) = det(A) | + | |
| + | <br> | ||
| + | <br> | ||
| − | Category:MA265Spring2010Walther | + | <br> Category:MA265Spring2010Walther |
Latest revision as of 15:14, 18 February 2010
Contents
Lecture 4 (01/21/10)
Row Echelon Form (ref)
Definition:Let A be a matrix, A will be a row echelon form(ref) if:
1. If any, a row full of zeros has to be at the bottom.
2. The leftmost nonzero in any row is "1", also known as 'leading 1's'
3. If row i and j are nonzero and i<j, the the 'leading 1' in row i is to the left of'leading 1'in row j
Note:Numbers following the 'leading 1's' can be any numbers
Reduced Row Echelon Form (rref)
Definition:Matrix A is in reduced row echelon form(rref) if:
1. A is in row echelon form(ref)
2. Any number above 'leading's 1' can only be zeros
Note:Every number in the column above the 'leading 1' need to be zeros
Elementary Transformation Steps for rref Conversion
1. Switching rows
2. Scale rows with any number
3.Take any row and add a scale version of any other row to it.
Note:It is always a good idea to try to rearrange rows to make the matrix easier to convert first and tries to avoid creating any fractions, especially in the early stages.
Properties of a Determinant
1. det(A) = det(transpose of A)
2. det(A with rows i and j interchanged) = -det(A)
3. det(A with row j replaced by row j +c*rowi) = det(A)
4. det(matrix with a row of zeros) = 0
5. det(matrix with 2 equal rows) = 0
- All of these properties stay true if you replace row with column
6. det(AB) = det(A)*det(B)
7. if det(A) = 0 then it has no inverse
8. det(BA(inverseB)) = det(A)
Category:MA265Spring2010Walther
