# Im(BF) in Im(B)?

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1. Mar 2, 2015

### Payam30

Hi,
Im actually doing some systems theory and it requires some basic linear algebra stuff that I totally forgotten. Anyway:

according to my prof. :
For any matrix F. Im(BF) is contained in Im(B).

here is my question:
so Im(FB) is still in Im(B)? or is it true only when we multiply a matrix to the right of the origin matrix?

if Im(B) is in Im(T) and Im(AB) is in Im(T), is Im(A) in Im(T) as well? and is Im(BA) in Im(T). imagin that the dimention ab matrices are appropriate. and they are constant real matrices

2. Mar 2, 2015

### Fredrik

Staff Emeritus
Assuming that B and F are matrices, and that Im(B) denotes the range of B (I would write it as ran B)...

If x is in the range of BF, then there's a y such that x=(BF)y=B(Fy). This implies that x is in the range of B.

So yes, Im(BF) is a subset of Im(B).

3. Mar 6, 2015

### Payam30

Hi
This is not necessarily true. For example, in $\mathbb{R}^2$, let $B$ a projection on the X-axis and let $B$ be a suitable rotation.
Not necessarily true. in $\mathbb{R}^2$, let $B$ and $T$ be projections on the X-axis and let $A$ be the identity.