Proving that S is a Basis for the n-Dimensional Vector Space V

[horefaen]

1
Suppose S is a set of n linearly independet in the n-dimensional vectorspace V. Prove that S is a basis for V.

My try at this proof is:
For S to be a basis for V it has to span V and the vectors in S needs to be linearly independent. But they have allreade sad that the vectors in S are linearly independent, so we only needs to show that it spans V.

But since V is n-dimensional it means that n linearly independent vectors in V span it, hence since S has n linearly independent vectors and is in V, S is a basis for V? Is this right?

2
Suppose that S is a set if n vectors that span the n-dimensional vector space V. Prove that S is a basis for V.

Now we need to show that the vectors in S is lineraly independent, right? But since n is n-dimensional it means that n lineraly independent vectors span it, since S is a set of vectors that spans V and S has n vectors, S is lineraly independent? Have I proved this one right?

 

[mighty2000]

Your proof is correct for both statements. To prove that S is a basis for V, you need to show that S spans V and that the vectors in S are linearly independent. Since S has n linearly independent vectors and V is n-dimensional, S must span V. And since S spans V and has n linearly independent vectors, it must also be a basis for V. Similarly, to prove that S is linearly independent, you need to show that S spans V and that the vectors in S are linearly independent. Since S spans V and has n linearly independent vectors, it must also be linearly independent.