1. Find Linearly independent or dependent vectors
`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`   `xx`    `3`   `4`   `4`   `5`

`+1`   `xx`    `-2`   `4`   `3`   `5`

`+1`   `xx`    `-2`   `3`   `3`   `4`

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

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2. Find Linearly independent or dependent vectors
`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`   `xx`    `2`   `3`   `4`   `5`

`-6`   `xx`    `0`   `3`   `3`   `5`

`+1`   `xx`    `0`   `2`   `3`   `4`

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

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