Can a matrix transformation ever make a linearly dependent matrix linearly independent? For example, if...
Can a matrix transformation ever make a linearly dependent matrix linearly independent?
For example, if A is a linearly dependent matrix, and B any matrix, could BA ever come out to be linearly independent?
Answer & Explanation
No it can not:
A is linear dependent, so there exist not all equal to 0 with
where or: there is a non-zero vector with
For BA we have
so regardless of the choice of B the vector is a non-zero kernel vector for BA, so BA can not be linear independent.
Let the matrices A and B be given. Suppose the columns of A are the column vectors where i runs over the number of columns, let that number be k.
So, your question boils down to asking can B act on the set of linearly independent vectors such that the resulting set is linearly dependent.
The answer is yes, let B be the transformation that mixes the columns of A, for example, define and for