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Home > Matrix & Vector calculators > Row Space example
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26. Row Space example
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- Example `[[1,-2,0,3,-4],[3,2,8,1,4],[2,3,7,2,3],[-1,2,0,4,-3]]`
- Example `[[1,2,3,2],[3,0,1,8],[2,-2,-2,6]]`
- Example `[[3,-1,-1],[2,-2,1]]`
- Example `[[-2,2,6,0],[0,6,7,5],[1,5,4,5]]`
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Other related methods
- Transforming matrix to Row Echelon Form
- Transforming matrix to Reduced Row Echelon Form
- Rank of matrix
- Characteristic polynomial of matrix
- Eigenvalues
- Eigenvectors (Eigenspace)
- Triangular Matrix
- LU decomposition using Gauss Elimination method of matrix
- LU decomposition using Doolittle's method of matrix
- LU decomposition using Crout's method of matrix
- Diagonal Matrix
- Cholesky Decomposition
- QR Decomposition (Gram Schmidt Method)
- QR Decomposition (Householder Method)
- LQ Decomposition
- Pivots
- Singular Value Decomposition (SVD)
- Moore-Penrose Pseudoinverse
- Power Method for dominant eigenvalue
- Determinant by gaussian elimination
- Expanding determinant along row / column
- Determinants using montante (bareiss algorithm)
- Leibniz formula for determinant
- determinants using Sarrus Rule
- determinants using properties of determinants
- Row Space
- Column Space
- Null Space
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25. determinants using properties of determinants
(Previous method) | 2. Example `[[1,2,3,2],[3,0,1,8],[2,-2,-2,6]]`
(Next example) |
1. Example `[[1,-2,0,3,-4],[3,2,8,1,4],[2,3,7,2,3],[-1,2,0,4,-3]]`
1. Find Row Space ...
`[[1,-2,0,3,-4],[3,2,8,1,4],[2,3,7,2,3],[-1,2,0,4,-3]]`Solution: | `1` | `-2` | `0` | `3` | `-4` | | | `3` | `2` | `8` | `1` | `4` | | | `2` | `3` | `7` | `2` | `3` | | | `-1` | `2` | `0` | `4` | `-3` | |
Now, reduce the matrix to reduced row echelon form interchanging rows `R_1 harr R_2` | = | | `3` | `2` | `8` | `1` | `4` | | | `1` | `-2` | `0` | `3` | `-4` | | | `2` | `3` | `7` | `2` | `3` | | | `-1` | `2` | `0` | `4` | `-3` | |
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`R_1 larr R_1-:3` | = | | `1` | `2/3` | `8/3` | `1/3` | `4/3` | | | `1` | `-2` | `0` | `3` | `-4` | | | `2` | `3` | `7` | `2` | `3` | | | `-1` | `2` | `0` | `4` | `-3` | |
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`R_2 larr R_2- R_1` | = | | `1` | `2/3` | `8/3` | `1/3` | `4/3` | | | `0` | `-8/3` | `-8/3` | `8/3` | `-16/3` | | | `2` | `3` | `7` | `2` | `3` | | | `-1` | `2` | `0` | `4` | `-3` | |
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`R_3 larr R_3-2xx R_1` | = | | `1` | `2/3` | `8/3` | `1/3` | `4/3` | | | `0` | `-8/3` | `-8/3` | `8/3` | `-16/3` | | | `0` | `5/3` | `5/3` | `4/3` | `1/3` | | | `-1` | `2` | `0` | `4` | `-3` | |
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`R_4 larr R_4+ R_1` | = | | `1` | `2/3` | `8/3` | `1/3` | `4/3` | | | `0` | `-8/3` | `-8/3` | `8/3` | `-16/3` | | | `0` | `5/3` | `5/3` | `4/3` | `1/3` | | | `0` | `8/3` | `8/3` | `13/3` | `-5/3` | |
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`R_2 larr R_2xx(-3/8)` | = | | `1` | `2/3` | `8/3` | `1/3` | `4/3` | | | `0` | `1` | `1` | `-1` | `2` | | | `0` | `5/3` | `5/3` | `4/3` | `1/3` | | | `0` | `8/3` | `8/3` | `13/3` | `-5/3` | |
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`R_1 larr R_1-2/3xx R_2` | = | | `1` | `0` | `2` | `1` | `0` | | | `0` | `1` | `1` | `-1` | `2` | | | `0` | `5/3` | `5/3` | `4/3` | `1/3` | | | `0` | `8/3` | `8/3` | `13/3` | `-5/3` | |
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`R_3 larr R_3-5/3xx R_2` | = | | `1` | `0` | `2` | `1` | `0` | | | `0` | `1` | `1` | `-1` | `2` | | | `0` | `0` | `0` | `3` | `-3` | | | `0` | `8/3` | `8/3` | `13/3` | `-5/3` | |
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`R_4 larr R_4-8/3xx R_2` | = | | `1` | `0` | `2` | `1` | `0` | | | `0` | `1` | `1` | `-1` | `2` | | | `0` | `0` | `0` | `3` | `-3` | | | `0` | `0` | `0` | `7` | `-7` | |
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interchanging rows `R_3 harr R_4` | = | | `1` | `0` | `2` | `1` | `0` | | | `0` | `1` | `1` | `-1` | `2` | | | `0` | `0` | `0` | `7` | `-7` | | | `0` | `0` | `0` | `3` | `-3` | |
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`R_3 larr R_3-:7` | = | | `1` | `0` | `2` | `1` | `0` | | | `0` | `1` | `1` | `-1` | `2` | | | `0` | `0` | `0` | `1` | `-1` | | | `0` | `0` | `0` | `3` | `-3` | |
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`R_1 larr R_1- R_3` | = | | `1` | `0` | `2` | `0` | `1` | | | `0` | `1` | `1` | `-1` | `2` | | | `0` | `0` | `0` | `1` | `-1` | | | `0` | `0` | `0` | `3` | `-3` | |
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`R_2 larr R_2+ R_3` | = | | `1` | `0` | `2` | `0` | `1` | | | `0` | `1` | `1` | `0` | `1` | | | `0` | `0` | `0` | `1` | `-1` | | | `0` | `0` | `0` | `3` | `-3` | |
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`R_4 larr R_4-3xx R_3` | = | | `1` | `0` | `2` | `0` | `1` | | | `0` | `1` | `1` | `0` | `1` | | | `0` | `0` | `0` | `1` | `-1` | | | `0` | `0` | `0` | `0` | `0` | |
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The rank of a matrix is the number of non all-zeros rows `:. Rank = 3` Row Space : The nonzero rows in the reduced row-echelon form are a basis for the row space of the matrix `[[1,0,2,0,1]],` `[[0,1,1,0,1]],` `[[0,0,0,1,-1]]`
This material is intended as a summary. Use your textbook for detail explanation.
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25. determinants using properties of determinants
(Previous method) | 2. Example `[[1,2,3,2],[3,0,1,8],[2,-2,-2,6]]`
(Next example) |
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