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Home > Matrix & Vector calculators > Linearly dependent and linearly independent vectors example
21. Linearly dependent and linearly independent vectors example ( Enter your problem )
( Enter your problem )
  1. Example-1
  2. Example-2
 		Other related methods
  1. Addition/Subtraction of two vectors
  2. Scalar Multiplication of vectors
  3. Dot Product of two vectors
  4. Cross Product of two vectors
  5. Magnitude(length) of a vector
  6. Unit vector
  7. Direction cosines of a vector
  8. Component form of a vector given two points
  9. Angle between two vectors
  10. Vector projections of b onto a
  11. Orthogonal vectors
  12. Collinear vectors
  13. Coplanar vectors
  14. Scalar triple product
  15. Vector triple product
  16. Area of triangle determined by two vectors
  17. Area of parallelogram determined by two vectors
  18. Volume of pyramid determined by vectors
  19. Volume of Parallelepiped determined by vectors
  20. Decomposition of vector in basis
  21. Linearly dependent and linearly independent vectors
1. Example-1
(Previous example)

2. Example-2


1. Find isLinearlyDependant(A,B,C)
`A=(5,-1,1)`,`B=(-2,3,4)`,`C=(3,4,5)`

Solution:
Here `vec A=(5,-1,1),vec B=(-2,3,4),vec C=(3,4,5)`


The vectors A,B,C are linearly dependent, if their determinant is zero. i.e. `|D|=0`

`|D|` = 
 `5`  `-1`  `1` 
 `-2`  `3`  `4` 
 `3`  `4`  `5` 


 =
 `5` × 
 `3`  `4` 
 `4`  `5` 
 `+1` × 
 `-2`  `4` 
 `3`  `5` 
 `+1` × 
 `-2`  `3` 
 `3`  `4` 


`=5 xx (3 × 5 - 4 × 4) +1 xx (-2 × 5 - 4 × 3) +1 xx (-2 × 4 - 3 × 3)`

`=5 xx (15 -16) +1 xx (-10 -12) +1 xx (-8 -9)`

`=5 xx (-1) +1 xx (-22) +1 xx (-17)`

`= -5 -22 -17`

`=-44``!=0`


Since `|D|!=0`, So vectors A,B,C are linearly independent.
2. Find ISLINEARLYDEPENDANT(A,B,C)
`A=(5,6,1)`,`B=(0,2,3)`,`C=(3,4,5)`

Solution:
Here `vec A=(5,6,1),vec B=(0,2,3),vec C=(3,4,5)`


The vectors A,B,C are linearly dependent, if their determinant is zero. i.e. `|D|=0`

`|D|` = 
 `5`  `6`  `1` 
 `0`  `2`  `3` 
 `3`  `4`  `5` 


 =
 `5` × 
 `2`  `3` 
 `4`  `5` 
 `-6` × 
 `0`  `3` 
 `3`  `5` 
 `+1` × 
 `0`  `2` 
 `3`  `4` 


`=5 xx (2 × 5 - 3 × 4) -6 xx (0 × 5 - 3 × 3) +1 xx (0 × 4 - 2 × 3)`

`=5 xx (10 -12) -6 xx (0 -9) +1 xx (0 -6)`

`=5 xx (-2) -6 xx (-9) +1 xx (-6)`

`= -10 +54 -6`

`=38``!=0`


Since `|D|!=0`, So vectors A,B,C are linearly independent.

This material is intended as a summary. Use your textbook for detail explanation.
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1. Example-1
(Previous example)


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