Made easy books GATE - 2015( IN ) 1 mark

Question

Let $A$ be a $n\times n$ matrix with rank $ r ( 0 < r < n ) .$Then $AX = 0$ has $p$ independent solutions,where $p$ is

$A)$  $r$            $B)$  $n$          $C)$  $n - r  $     $D)$  $n + r$

Comments

not sure, but may be c is the correct Choice?

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yes it will be n-r.

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please explain

what mean by independent solution?

Answer 1

ans : ( c ) n-r

explanation :

As rank is r, which is lesser than n, there should be infinite solutions! but wait,
A matrix represents linear transformation of a vector space. when the rank is less than column space, it simply means that, after applying that transformation, all vectors in n dimensions are mapped to vectors in r  dimensions, (for example from 3d to 2d).

So, in this case, n dimension space got shrunk into r dimensions,  that means n-r dimensions  got vanished. a vector in any one of that n-r dimensions cannot be expressed as a linear combination of other dimensions (they are independent) but! all the infinite vectors in any of the n-r dimensions can be expressed as a linear combinations of other vectors in the same dimension.

therefore, there are n-r independent solutions

PS : If you think that you have no clue of what i am telling you, what you need is a geometrical understanding of linear algebra! if you are interested, watch https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

Simple example :
 

consider the matrix

1 0 0

0 0 0

0 0 0

(shrinks 3D space  to 1D)

now the independent solution sets are

0

y

0

(all vectors in y axis, put any thing to y ) and

0

0

z

(all the vectors in z axis, put any thing to z)

The infinite solutions which everyone says is all the vectors that lied in yz plane before applying this transformation!

that is

0

x

y

(that's your infinite solution set :-)