Let say K and L are two different subspace real vector space V. If given...

elbluffz1

elbluffz1

Answered

2022-01-24

Let say K and L are two different subspace real vector space V. If given dim(K)=dim(L)=4, how to determine minimal dimensions are possible for V?

Answer & Explanation

Madelyn Townsend

Madelyn Townsend

Expert

2022-01-25Added 13 answers

Explanation:
Let the four vectors k1,k2,k3 and k4 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 l1 in L, which is not in K, i.e, which is not a linear combination of k1,k2,k3 and k4.
So, the set {k1,k2,k3,k4,l1} 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 {k1,k2,k3,k4,l1} 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 R5 and let K and V consists of vectors of the forms
(αβγδ0) and (μνλ0ϕ)
It is easy to see that the vectors
(10000),(01000),(00100)and(00000)
form a basis of K Append the vector (00000) , and you will get a basis for the entire vector space,

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