Linear Algegra: Nul A and Col A relationship if Nul A is not the zero space

[imagenesis]

1. If R is a 6X6 matrix, and Nul R is not the zero space, what can you say about Col R?





3. Well we know that there is some vector that can be added to R to form 0... But how does that relate to Col R ? I mean I really don't know. I am guessing it has something to do with the m where mxn is the size of R.

 

[Vid]

Look at the rank-nullity theorem.

 

[HallsofIvy]

If R is a 6 by 6 matrix- i.e. represents a linear transformation from 6 dimensional vector space to a 6 dimensional vector space, then the dim of the null space of R plus the dimension of the column space of R is equal to 6- that the rank-nullity theorem Vid mentioned. If the nullity of R is greater than 0, what can you way about the rank of R?