Let say K and L are two different subspace real vector space V. If given...
Let say K and L are two different subspace real vector space
V. If given , how to determine minimal dimensions are possible for V?
Answer & Explanation
Let the four vectors and form a basis of the vector space K. Since K is a subspace of V, these four vectors form a linearly independent set in V. Since L is a subspace of V different from K, there must be at least one element, say in L, which is not in K, i.e, which is not a linear combination of and .
So, the set s a linear independent et of vectors in V. Thus the dimensionality of V is at least 5!
In fact, it is possible for the span of to be the entire vector space V - so that the minimum number of basis vectors must be 5
Just as an example, let V be and let K and V consists of vectors of the forms
It is easy to see that the vectors
form a basis of K Append the vector , and you will get a basis for the entire vector space,