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?

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

Expert

2022-02-01Added 11 answers

Step 1

No it can not:

$A\in {K}^{m\times n},B\in {K}^{p\times m}$

A is linear dependent, so there exist${\lambda}_{k}\in K$ not all equal to 0 with

$\sum _{k=1}^{n}{\lambda}_{k}{a}_{k}=0.$

where$A=({a}_{1},{a}_{2},\dots ,{a}_{n})$ or: there is a non-zero vector ${x}_{0}=\left({\lambda}_{k}\right)\ne 0$ with

$A{x}_{0}=0.$

For BA we have

$\left(BA\right){x}_{0}=B\left(A{x}_{0}\right)=B0=0$

so regardless of the choice of B the vector$x}_{0$ is a non-zero kernel vector for BA, so BA can not be linear independent.

No it can not:

A is linear dependent, so there exist

where

For BA we have

so regardless of the choice of B the vector

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$A}_{i$ where i runs over the number of columns, let that number be k.

Now,

$BA=[B{A}_{1}\left|B{A}_{2}\right|\dots B{A}_{k}]$

So, your question boils down to asking can B act on the set of linearly independent vectors$\{{A}_{1},{A}_{2},\dots {A}_{k}\}$ such that the resulting set $\{B{A}_{1},B{A}_{2},\dots B{A}_{k}\}$ is linearly dependent.

The answer is yes, let B be the transformation that mixes the columns of A, for example, define$B{A}_{1}={A}_{1}+{A}_{2}$ and $B{A}_{i}={A}_{i}$ for $i\ge 2$

Let the matrices A and B be given. Suppose the columns of A are the column vectors

Now,

So, your question boils down to asking can B act on the set of linearly independent vectors

The answer is yes, let B be the transformation that mixes the columns of A, for example, define

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