# Can I have a detailed explanation on the following equivalent statement

proof, according to the instructions given after the question?

The proof should be suitable for college level linear algebra module and the explanation should be easy-to-follow.

Question :

Let A be an n*n matrix.

Prove that the following statements are equivalent.

(a) The row vectors of A are linearly independent.

(b) The columns vectors of A span R^n.

(c) The row vectors of A span R^n.

(d) The columns vectors of A form a basis for R^n.

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Instructions provided :

When proving statements are equivalent it should be done in the following order.

(1) Start with the fact that A is an n*n matrix

(2) Prove (a)==>(b)

(3) Prove (b)==>(c)

(4) Prove (c)==>(d)

( Prove (a)==>(b) means to assume (a) is true and then proving (b) is true, using that )

The proof will be considered incorrect, if the statements are not proved as in above order and above method.

Thank you!

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