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Home > Matrix & Vector calculators > Null Space example (Nullity of a matrix)
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24. Null Space example
( Enter your problem )
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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
- determinants using Sarrus Rule
- determinants using properties of determinants
- Row Space
- Column Space
- Null Space
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2. Example `[[1,2,3,2],[3,0,1,8],[2,-2,-2,6]]`
(Previous example) | 4. Example `[[-2,2,6,0],[0,6,7,5],[1,5,4,5]]`
(Next example) |
3. Example `[[3,-1,-1],[2,-2,1]]`
Find Null Space ...
`[[3,-1,-1],[2,-2,1]]`
Solution:
Now, reduce the matrix to reduced row echelon form
`R_1 larr R_1-:3`
| = | | `1` | `-1/3` | `-1/3` | | | `2` | `-2` | `1` | |
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`R_2 larr R_2-2xx R_1`
| = | | `1` | `-1/3` | `-1/3` | | | `0` | `-4/3` | `5/3` | |
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`R_2 larr R_2xx-3/4`
| = | | `1` | `-1/3` | `-1/3` | | | `0` | `1` | `-5/4` | |
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`R_1 larr R_1+1/3xx R_2`
The rank of a matrix is the number of non all-zeros rows
`:. Rank = 2`
Null Space :
Now, solve the matrix equation
`x_1-3/4x_3=0`
`x_2-5/4x_3=0`
Add equation for each free variable
`x_1-3/4x_3=0`
`x_2-5/4x_3=0`
`x_3=x_3`
Solve for each variable in terms of the free variables
`x_1=3/4x_3`
`x_2=5/4x_3`
`x_3=x_3`
Convert this into vectors
| = | | = | `[[3/4],[5/4],[1]]` | `x_3` |
Thus, the basis for the null space is
`[[3/4],[5/4],[1]]`
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
2. Example `[[1,2,3,2],[3,0,1,8],[2,-2,-2,6]]`
(Previous example) | 4. Example `[[-2,2,6,0],[0,6,7,5],[1,5,4,5]]`
(Next example) |
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