ZEAL TEST

Question

Isn’t the “No of independent rows”= RAnk of matrix?

Comments

Yes true .But the question asked number of independent solution means the solution of $AX=0$ has how many independent vectors not the given matrix A has how many independent solution.

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@Kabir5454 Please elaborate how independent solutions and Rank of matrix are related. It seems I am not able to correlate.

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@DAWID15 I recommend you to watch GO class free youtube video of Linear Algebra. Your doubts will clear.

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Remind Rank-Nullity theorem,

No. of Columns OR No. of Variables = Rank + Nullity.

Here,
Rank = No. of Independent Columns OR No. of Pivot Variables.
Nullity = No. of Free Variables OR No. of Independent Solutions.

So,
1000 = 640 + Nullity
Nullity = 360

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A follow up to @DebSujit ‘s comment

IF they ask for linearly independent solutions in solution of Ax=b => they are asking about the number of free variables

IF they ask for linearly independent rows/colums in A in Ax=b => they are asking about the number of pivots(or Rank)

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@Shubhodeep Correct!

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@DebRC free and independent variable are same.

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@samarpita fixed

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if rank of the matrix is R and  N is the order of the matrix then

in this matrix N-R indpendent  vector  exist or independent solution exist

1000-640=360

Answer 1

Number of Linearly Independent columns = Rank

Number of Linearly Independent solutions = Nullity

 

Here n = 1000

rank = 640

Therefore number of free columns/ nullity = 1000-640 = 360 = Number of independent solutions.