2008 dse doubt

[Ansh malhotra]

[Ansh malhotra]

[Ansh malhotra]

Please help me in above question

[Nikkita]

The solution exists in case determinant is not equal to zero. Ans- (a)

[Mike]

answer for 1 should be (d)

[Ansh malhotra]

Please mike explain your answer

[knowpraveen]

We are not given that m=n, in which case we could have easily gone with detA not equal to 0. Since m could be greater than or less than or equal to n, let's take these cases one by one. If m>n, it would be mean number of equations is greater than number of variables, which means there is no solution. Take an example to confirm.

For m<n, there are solutions because since one variable can choose an arbitrary value and others follow.

Column independence is out of question. We are left with options, det A not equal to 0 and the row independence case. For det A not equal to 0, row independence is a necessary condition for (m=n), otherwise one row becomes nullified and we wouldn't be able to solve it any further. Hence, we zero in on row independence which is consistent with our idea of homogenous and inhomogenous linear equations.

[knowpraveen]

g is linear, f is not. g satisfies homogeniety and additive property, f doesnt.

[Ansh malhotra]

Thank you so much

[knowpraveen]

Anytime bro!

[Ansh malhotra]

I got homogeneity property what is additive property

[knowpraveen]

The additive property is, f((0,0)+(0,0))=f(0,0)+f(0,0) i.e f(0,0) is not equal to 2f(0,0) as you will see by substituting in the transformation.

But, it satisfies in the case of g.