Question

$q_1,q_2,q_3...q_m$ are n-dimensional vectors with m<n. This set of vectors is linearly dependent. Q is the matrix with $q_1{q}_2,q_3...q_m$ as the columns. the rank of Q is

• A. less than n
• B. m
• C. between m and n
• D. n

Answer 1

The correct option is A less than n

Given Q=[$q_1{q}_2{q}_3...q_m$ ]${}_{n\times m}$ and m < n.
(By property of Rank ) we have $\rho \left(Q\right)\le m$
But given that $q_1{q}_2{q}_3...q_m$ are dependent vectors $\rho \left(Q\right)\le m$ (No. of vectors)

$\therefore$ m < n so we can conclude that $\rho \left(Q\right)<n$ also.