Show that in an n-dimensional linear space we can find at most n linearly independent elements.
The solution is m=n
If a linear space V has a basis with n elements, then all other bases of V consist of n elements as well.
We say that n is the dimension of V that is dim(V)=n.
Consider two basis of V for a n-dimensional spaces as follows.
To show that n is the largest possible dimension for an n-dimensional space that m=n.
Since, the m vectors of form a basis.
We know that as follows.
, By the definition of linear independence.
Also know that forma a basis for the same space with n vectors and since the dimension of the space can’t change and as follows.
Since, m=n, that is there is no such basis m exists where for an n-dimensional space.
Thus, n is the largest possible dimension for an n-dimensional space that m=n
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