Linear independence of stacked vectors. Consider the stacked vectors where a1, . . . , ak are…

Linear independence of stacked vectors. Consider the stacked
vectors

where a1, . . . , ak are n-vectors and b1 , . . . , bk are
m-vectors

(a) Suppose a1 , . . . , ak are linearly independent. (We
make no assumptions about the vectors b1, . . . , bk.) Can we conclude that the
stacked vectors c1 , . . . , ck are linearly independent?

(b) Now suppose that a1, . . . , ak are linearly dependent.
(Again, with no assumptions about b1, . . . , bk.) Can we conclude that the
stacked vectors c1 , . . . , ck are linearly dependent?