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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Answered: (3) (a) Let T be a surjective linear…

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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linear algebra

(3) (a) Let T be a surjective linear transformation from P 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(P,) 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 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. No
justification is needed beyond what is described below.
(i) If u, v, w are linear dependent, then w is a linear combination of
u and v.
• If true, state a general theorem or statement for which this
is 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.

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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