union of two linear independent set

Question

I have two linearly independent sets in $R^n$ space.One with cardinality $n-k$ and another with cardinality $k+1$.Is the union of these two sets linearly independent?Actually I came across this problem while studying the proof of young-eckart theorem for low rank approxiamtion of matrix.They say that it cant thats why their intersection of their span is non null.but I cant find out why it is so.

Answer 1

It depends on what you mean by union. If you consider the set theoretic union, then you can get both a linearly dependent or linearly independent set. An instance of the second case is $$ A=\{e_1,e_2\},\quad B=\{e_2,e_3\} $$ in $\mathbb{R}^3$, where the vectors are the ones in the canonical basis.

If the sets are disjoint, you end up with a set having $n+1$ vectors, which is certainly linearly dependent. If the set are not disjoint, you simply can't decide without examining the vectors. Example: $$ A=\{e_1,e_2\},\quad B=\{e_2,e_1+e_2\},\quad A\cup B=\{e_1,e_2,e_2+e_2\} $$ and the union is linearly dependent.

If, instead, you consider lists of vectors rather than sets (somebody calls them ordered sets of vectors) and the union is juxtaposition of lists, then your union is surely not linearly independent, because no list of $n+1$ vectors in an $n$-dimensional vector space is linearly independent. I guess your textbook is using this notion of “set”, when talking of linear dependence.

Answer 2

The union is obviously not linearly independent: The cardinality of the union is $n+1>n$ if the sets are disjoint, and any set of cardinality $> n$ in $\mathbb{R}^n$ is linearly dependent.

On the other hand, if the sets are not disjoint, then we are obviously done.