
#1
Mar1612, 12:30 AM

P: 3

Hi all,
I am trying to find the probability that a matrix has full rank. Consider a K*N matrix where the first K columns are linearly independent columns and the next NK columns are linear combinations of these K columns. I want to find the probability that a sub matrix formed by randomly selecting columns of this matrix has full rank. (or all the columns of this sub matrix are linearly independent). My logic is as follows, Step 1 : Select u1 number of columns randomly from the first K columns. Then rank(Gu) = u1. No. of ways to select = K choose u1 Step 2: Now I select one column from the NK columns and check whether this belong to the span of u1 columns. If not then I increase rank by one. span of u1 contain 2^u1 possibilities. So ideally I have to select 1 from 2^K  2^u1 columns in order to have rank(Gu) = u1 + 1 But my problem is that, NK < 2^K so the total number of columns I have to make the selection is NK and not 2^K. I'm finding it really difficult to interpret this in mathematical formulas using combinations. Really appreciate if someone can help. Thanks in advance. 


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