1. Find Linearly independent or dependent vectors
`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`   `xx`    `5`   `6`   `8`   `9`

`-2`   `xx`    `4`   `6`   `7`   `9`

`+3`   `xx`    `4`   `5`   `7`   `8`

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

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2. Find Linearly independent or dependent vectors
`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`   `xx`    `4`   `6`   `4`   `5`

`-2`   `xx`    `2`   `6`   `3`   `5`

`+3`   `xx`    `2`   `4`   `3`   `4`

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

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