Is it true these are theorems in Linear Algebra?

[Logan Land]

If each vector in basis B1 is scalar multiple of some vector in basis B2 then transition matrix PB1→B2 is diagonal.The column space of matrix A is the set of solutions of Ax = b.If A is n × n invertible matrix and AB is defined then row space of AB coincides with row space of B.Column space of skew symmetric matrix coincides with its row space.If A and B are n×n matrices that have the same row space, then A and B have the same column space.

 

[caffeinemachine]

If each vector in basis B1 is scalar multiple of some vector in basis B2 then transition matrix PB1→B2 is diagonal.The column space of matrix A is the set of solutions of Ax = b.If A is n × n invertible matrix and AB is defined then row space of AB coincides with row space of B.Column space of skew symmetric matrix coincides with its row space.If A and B are n×n matrices that have the same row space, then A and B have the same column space.

Hello Lland314,

Can you please post some details on your attempt on these and where exactly you are stuck?

Also, try not posting more than 2 questions in a single thread.

 

[Shaakasu808]

If each vector in basis B1 is scalar multiple of some vector in basis B2 then transition matrix PB1→B2 is diagonal.If A and B are n×n matrices that have the same row space, then A and B have the same column space.

Actually I am having issue attempting to prove the first quoted question as true, as such i believe it is false. But intuitively I think it is true.

As for the second quoted question the answer I am unsure if he meant an mxn or actually an nxn matrix