Are the Vectors S and T in the column of (ABC)

[TomSavage]

I have no clue how to decode this question or do it but I was given the vectors S=[1 1 0] and T=[-1 0 1] and asked to determine whether or not they are in the column space of ABC when A, B and C are 3x3 matrices. My prof hinted to "think of rank and nullity", can someone please point me in the direction of the question cause as it stands right now I am lost, thanks.

 

[I like Serena]

Hi Tom,

Hint 1: A vector is in the column space if it is a linear combination of the columns of the matrix.
Perhaps we can already visually see it.

Hint 2: We can find the rank of a matrix by Gaussian elimination and see how many vectors remain. Now add the vector in question, and repeat the elimination process. If we are left with an additional vector, it must have been independent of the original set, and is therefore not in the column space. Otherwise it was dependent, and is in the column space.

Hint 3: If we already have a full rank set of vectors, adding a vector cannot increase the rank. Then it must be dependent and in the column space.

Hint 4: If we know the rank of the null space, we can deduce the rank of the column space by the rank-nullity-theorem. That may help for hint 3 and hint 2.