Column Spaces

[loli12]

can anyone help me with this?

Let A be in R^mxn, B in R^nxr, and C=AB. Show that:
(i) The column space of C is a subspace of the column space of A;
(ii) Rank(C) is smaller than or equal to min{rank(A), rank(B)}.

For (i) I tried to show that C can be written as linear combination of A but seems like I am missing something....

Please help!! Thanks

[shmoe]

For (i) you want to show each column of C is a linear combination of the column's of A. How would you find the first column of C?

For (ii), rank(C) is the dimension of it's column space, so using part (i), what can you say about it compared to rank(A)? To compare it to rank(B), you'll need the row space analogue of part (i), "the row space of C is a subspace of the row space of B". You can get this from part (i) with little effort by taking transposes, which handily converts a question about rows to a question about columns.