GO Classes CS/DA 2025 | Weekly Quiz 4 | Linear Algebra | Question: 29

Answer 1

$\mathrm{S} 1$ : In this case, there is always a solution. It will always have a unique solution iff cols of matrix are LI. Hence, $\mathrm{S} 1$ is true.

$\text{S2} :$ In this case, if the the column vectors cover $b$ in their subspace, then their will be infinitely many solutions(as multiple linear combinations can satisfy $b$ since, all the col vectors are LD), else, no solution case will happen. Hence, $\text{S2}$ is false.

It is based on an imp statement in lec :

If a vector can be represented as a lin comb of few vectors and further if those few vectors are LI then there is just one way to do it, otherwise there are many ways.

Comments

Answer will be c
as S1 is true and S2 is False

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for S2: we can also explain like
if columns of matrix are dependent then the matrix is definitely  not a full rank matrix
So there exist free var in echelon form , hence infi soln exist.

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Please mention that if soluition exits then there will be infinit many solution
i.e. either 0 or infinite

Answer 2

Here the answer is option :C. 

Case 1:  for S1 there it is said that b is Linear combination of column of A so there will be solution

now weather the solution is unique or whether it is infinite it depends on  Columns of A are linear independent or linearly dependent 

IF the columns are L.I then there is a unique solution and if they are L.D there is infinite solution

so S1 is true 

 

​​​​​​CASE 2 : for S2 it is said that b is linear dependent on Columns of A and there is a unique solution whic is false because if b is L.D then there is always infinite solution