@amistre64 @Loser66
Please explain the relationship between dimensions of a matrix's column space, null space, row space ( column space trasposed) and the null space transposed

Question

@amistre64 @Loser66 
Please explain the relationship between dimensions of a matrix's column space, null space, row space ( column space trasposed) and the null space transposed

anonymous · Answer

The row space of a Matrix A is the orthogonal complement of the null space of that Matrix. Therefore they are linearly independent and span the entire room R^n

Same argumentation for your remaining Subspaces.

caozeyuan · Answer

no, no,no! I remember my teacher told us that if the matrix A is m by n then the column space is in R^n while it's row space is in R^n. ANd he talked about number of pivot columns, number of free variable, things like that. But I can't recall the final conclusion

anonymous · Answer

Well in your question it asks for a "relationship between dimensions" of a matrix0s column space, null space, row space and the left null space (which is the null space transposed), if you don't require to argument through orthogonal complements then just do it as you know:

rank(A) = dim(rowsp(A)) = dim(colsp(A)),
rank(A) = number of pivots in any echelon form of A,
rank(A) = the maximum number of linearly independent rows or columns of A.

Maybe this picture will help you a bit too:

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