Sachin Mittal Sir's course on GO Classes

Question

It is given that m < n
lets consider m = 4 and n = 5.
If rank(A) = n, this means no of linearly independent columns in A are 5, but we cannot have 5 linearly independent vectors in R4.
Where is the problem ?

Comments

if A is m*n matrix then rank(A) $\leq$ min(m,n)

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Exactly rank(A)<= min(m,n)

Answer 1

Lets consider a small matrix of size m * n such that m<n where m=2 and n=3
Rank of A is given as n which means we have n pivots,which is impossible .Lets understand by the diagram

Here we can have atmost 2 pivots,Rank of the matrix can be 1 or 2 or 0(only for NULL matrix) so the statement u made is wrong as rank cannot be n ...that’s why the conclusion u r deriving is true that we cannot have more than n linearly independent vectors in R$^{n}$ space

Answer 2

Given m<n for given matrix A

In $R^{m}$, Atmost m linearly independent vectors are possible .

Number of linear independent vectors must be $\leq$ m

Answer 3

In mXn matrix if m<n then rank will never be n ,

Because we can form maximum pivot elements m .

And , in matrix of space R4 there will at most 4 linearly independent vectors can be formed, because that 4 vector will form whole space in R4