The Rank+Nullity Theorem
نویسنده
چکیده
The rank+nullity theorem states that, if T is a linear transformation from a finite-dimensional vector space V to a finite-dimensional vector space W , then dim(V ) = rank(T ) + nullity(T ), where rank(T ) = dim(im(T )) and nullity(T ) = dim(ker(T )). The proof treated here is standard; see, for example, [14]: take a basis A of ker(T ) and extend it to a basis B of V , and then show that dim(im(T )) is equal to |B −A|, and that T is one-to-one on B −A.
منابع مشابه
Math 312 Final Exam
a) What are the possible values for the rank of A? Why? Solution By the Rank-Nullity Theorem 0 ≤ rank (A) ≤ min{6, 4} = 4. b) What are the possible values for the dimension of the kernel of A? Why? Solution Since dim(image(A)) ≤ 4, by the Rank-Nullity Theorem 2 ≤ dim ker(A) ≤ 6. c) Suppose the rank of A is as large as possible. What is the dimension of ker(A)? Explain. Solution Since then dim(i...
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(c) Columns of A are independent. (d) A is tall (i.e., n ≤ m) and full-rank (i.e., rank(A) = min(m,n) = n). Solution: We will show the chain of equivalences (a) =⇒ (b) =⇒ (c) =⇒ (d) =⇒ (a). (a) =⇒ (b): By the rank–nullity theorem, we have dim(N (A)) + rank(A) = n, which implies rank(A) = n (since dim(N (A)) = 0). Since rank(A) = rank(A ), we then have rank(A ) = n. Since rank is equivalent to t...
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