linear algebra (3) (a) Let T be a surjective linear transformation from P W. Show thatdim(W) <8. Complete the following "proof":(Note: If two blanks share the same letter, then they must be the sameanswer.)Proof. By the Rank-Nullity Theorem, we know that dim(P7), rank(T),and nullity (T) are related by the equationall non-negative numbers, we know that dim(P,) must berank(T).By definition, rank(T) is the dimension of theever, since T is surjective, thehave that dim(P7) > dim(W).Lastly, the dimension of P, is(A)Since these are(В)(C)of T. How-(C)of T is(D)So, we(E)Thus dim(W)(E) (b) Determine if each of the following statements is true or false. Nojustification is needed beyond what is described below.(i) If u, v, w are linear dependent, then w is a linear combination ofu and v.• If true, state a general theorem or statement for which thisis an example.• If false, find a counterexample.(ii) The set| 2² + y = 1 is a subspace of R.• If true, show that it is.a subspace.• If false, find a counterexample to one of the subspace prop-erties.

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(3) (a) Let T be a surjective linear transformation from P7 W. Show that dim(W) < 8. Complete the following "proof": (Note: If two blanks share the same letter, then they must be the same answer.) Proof. By the Rank-Nullity Theorem, we know that dim(P7), rank(T), and nullity(T) are related by the equation all non-negative numbers, we know that dim(P7) must be rank(T). By definition, rank(T) is the dimension of the ever, since T is surjective, the have that dim(P7) > dim(W). Lastly, the dimension of P7 is (A) Since these are (B) (C) of T. How- (C) of T is (D) So, we (E) Thus dim(W) < (E)

(3) (a) Let T be a surjective linear transformation from P7 W. Show that dim(W) < 8. Complete the following "proof": (Note: If two blanks share the same letter, then they must be the same answer.) Proof. By the Rank-Nullity Theorem, we know that dim(P7), rank(T), and nullity(T) are related by the equation all non-negative numbers, we know that dim(P7) must be rank(T). By definition, rank(T) is the dimension of the ever, since T is surjective, the have that dim(P7) > dim(W). Lastly, the dimension of P7 is (A) Since these are (B) (C) of T. How- (C) of T is (D) So, we (E) Thus dim(W) < (E)

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

Problem 4EQ

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