Linear algebra proof?
I need help with these two questions.
1. If A is an mxn matrix and B1, B2,…,Bk are columns in Rm such that the system AX=Bi has a solution Xi for each i=1,…,k. Show that {B1, B2,…,Bk} is independent in Rm then {X1,X2,…,Xk} is independent in Rn
2. If A,B are matrices and the columns of AB are independent, show that the columns of B are independent.
Any help would be much appreciated.
Comments
1. I assume you mean "AX=Bi has a *nontrivial* solution Xi...".
Suppose the {Xi} are nonzero and dependent. That means that there are constants ci, not all zero, such that:
0 = Σ ci*Xi ... then multiply on the left by A
0 = A Σ ci*Xi = Σci*AXi = ΣciBi
That proves that if {Xi} is dependent then {Bi} is dependent. The contrapositive of is also true: If {Bi} is independent, then {Xi} is independent.
2. Let C be a nonzero column vector with entries ci. The product BC is simply a linear combination of the columns of B with ci as the coefficients, so a restatement of dependence is that the columns of B are dependent iff there exists a nonzero C such that BC=0. Assume that's true and multiply on the left by A (sound familiar?):
0 = BC
0 = A(BC) = (AB)C
...and the columns of AB are dependent. If the columns of B are dependent then the columns of AB are too. The contrapositive is, again, also true: If the columns of AB are independent then so are the columns of B.