Probability of a matrix having full rank
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 N-K 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 N-K 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, N-K < 2^K so the total number of columns I have to make the selection is N-K 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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