Linear dependence dse 2011

[aditi5000]

Q 4 from DSE 2011
we are told that x solves the system of equations Ax=b. A is m*n matrix. Then we are asked {a1,a2,a3....,an,b} is ...
1. linearly indep
2. linearly dep
3. linearly dep only if a1,...an are linearly dep
4. linearly dep if m=n

Please tell me how the answer is 2. If Ax=b is solved by x means determinant A =/= 0 and also rank of A and augmented A is the same. and i think {a1,a2...an,b} is the augmented matrix right? and rank indicates no of linearly independent rows. so then shouldnt the answer be 1... is it  2 because  linear dependence means non-trivial solutions? and we have been told that a solution exists...? Please give the logic behind this answer. would really appreciate it.

[Chinni18]

Hey aditi, consider this:
2x+3y= 8
x+5y=11
The very fact that x and y are able to solve this set of equations implies that the vectors [2,1], [3,5] and [8,11] are linearly dependent. Because a linear combination of the first two gives you the third one.
On the other hand, if the questions asks only about {a1,a2,a3,..,an} and not {a1,a2,a3,..,an,b}, then in that case your answer should be a, ie linearly independent.

[ritu]

hi aditi i will tell u what i feel nd then see wot others say...
in this case no doubt the vectors {v1,v2,........,vn} are independent nd that why they can form a liner combination "with" each other to give rise to "b " vector....since there exists a relation btwn the vi vectors and b vector it means vi and b are dependent coz b lies in the n dimensional space generatef by vi vectors....had the solution x not existed then vi and b wud have been independent....
lets c if this logic is correct or nt:-)

[Chinni18]

I used the same logic Ritu  I think it is correct. Intuitively it makes a lot of sense.

[aditi5000]

Hey Chinni and Ritu, thanks! that makes a lot of sense now that you are saying it in this manner :) but tell me how did you eliminate option c and d? they also mention linear dependence but additionally give some fuzzy conditions.

[Chinni18]

Well aditi if c is true then the columns in the matrix A are linearly dependent which yields a trivial solution and holds no relationship with b at all. So c has to be eliminated.
And d also neddn't always be true, becaue there are many questions where m and n are not equal and yet we get a solution. A doesn't need to be a square matrix.

[aditi5000]

Ah yes.... you are right. Perfect, thank you so much!!!!!

[Chinni18]

You are welcome aditi