Can a matrix transformation ever make a linearly dependent matrix linearly independent? For example, if...

FiessyFrimatsd0

FiessyFrimatsd0

Answered

2022-01-31

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

Ronald Alvarez

Ronald Alvarez

Expert

2022-02-01Added 11 answers

Step 1
No it can not:
AKm×n,BKp×m
A is linear dependent, so there exist λkK not all equal to 0 with
k=1nλkak=0.
where A=(a1,a2,,an) or: there is a non-zero vector x0=(λk)0 with
Ax0=0.
For BA we have
(BA)x0=B(Ax0)=B0=0
so regardless of the choice of B the vector x0 is a non-zero kernel vector for BA, so BA can not be linear independent.
Lily Thornton

Lily Thornton

Expert

2022-02-02Added 19 answers

Step 1
Let the matrices A and B be given. Suppose the columns of A are the column vectors Ai where i runs over the number of columns, let that number be k.
Now,
BA=[BA1|BA2|BAk]
So, your question boils down to asking can B act on the set of linearly independent vectors {A1,A2,Ak} such that the resulting set {BA1,BA2,BAk} is linearly dependent.
The answer is yes, let B be the transformation that mixes the columns of A, for example, define BA1=A1+A2 and BAi=Ai for i2

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