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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
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  20. Decomposition of vector in basis
  21. Linearly dependent and linearly independent vectors
20. Decomposition of vector in basis
(Previous method)		2. Example-2
(Next example)

1. Example-1


1. Find isLinearlyDependant(A,B,C)
`A=(1,2,3)`,`B=(4,5,6)`,`C=(7,8,9)`

Solution:
Here `vec A=(1,2,3),vec B=(4,5,6),vec C=(7,8,9)`

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

`|D|` = 
 `1`  `2`  `3` 
 `4`  `5`  `6` 
 `7`  `8`  `9` 


 =
 `1` × 
 `5`  `6` 
 `8`  `9` 
 `-2` × 
 `4`  `6` 
 `7`  `9` 
 `+3` × 
 `4`  `5` 
 `7`  `8` 


`=1 xx (5 × 9 - 6 × 8) -2 xx (4 × 9 - 6 × 7) +3 xx (4 × 8 - 5 × 7)`

`=1 xx (45 -48) -2 xx (36 -42) +3 xx (32 -35)`

`=1 xx (-3) -2 xx (-6) +3 xx (-3)`

`= -3 +12 -9`

`=0`


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

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


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

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


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


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

`=1 xx (20 -24) -2 xx (10 -18) +3 xx (8 -12)`

`=1 xx (-4) -2 xx (-8) +3 xx (-4)`

`= -4 +16 -12`

`=0`


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

This material is intended as a summary. Use your textbook for detail explanation.
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20. Decomposition of vector in basis
(Previous method)		2. Example-2
(Next example)


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