8. Let V and W be the vector spaces and T : V → W be linear. (a) Prove that T is one-to-one if and only if T carries linearly independent subsets of V onto linearly independent subsets of W. (b) Suppose that T is one-to-one and that S is a subset of V. Prove that S is linearly independent if and only if T(S) is linearly independent. (c) Suppose 3 = {v1, t2, ., v,} is a basis for V and T is one-to-one and onto. Prove that T(3) = {T(v1), T(v2), .. ,T(v,)} is a basis for W.

Q: VnEN+, 5" + 3n > 22n+1

A: Given that the statement ∀n∈ℕ+ , 5n+3n≥22n+1. The objective is to prove the given statement.…

Q: Determine L-1) 9s – 8 (? – 2sJ*

A:

Q: For all integers a, b, and c, if a/b and a|c, then a|(5b – 7c). -

A:

Q: prove if P>-l then Citp = I+np %3D n€ IN

A: The given statement is not TRUE. Counter example is taking n = 2.

Q: 'rove 1 det(A) = %3D det (41)

A:

Q: t FCK and KCK' be algebraic extensions. If both FCK and

A:

Q: Prove that {i} = I. ¡El

A:

Q: Prove that.

A:

Q: 12 y 2 4xy³z² 4877

A: formula used  xa×xb=xa+blet  A=12x4y8z74xy3z2

Q: 2 Prove that | (0,1)| = |[0, 1]|. (Hint:

A:

Q: -, first prove that )" 6.

A:

Q: 7 (V3 +4} = - = - 64 (V3+i) 2) Using de Moiver's formula prove that:

A:

Q: 6.) Derike the binomial idendity nent) (un-) C Hint: For mzz, c^)+2(^) +m?]

A: To derive the binomial identity22+42+62+.........+2n2=nn+14n-16

Q: 5) Suppose that W1 < R" and W2 4 R". Prove (W1 n W2) < R"

A:

Q: Prove that.

A:

Q: 1 Show that n2 + 1 IN V

A:

Q: [2(c056'+isin6")ゴー sin6")ゴー %3D

A: Given:2cos 50+sin 506

Q: Prove bey wnduc hiou that for n22.

A:

Q: et x + y+ z=5 where x, y, z E R. Prove that x² + y² + z² >

A:

Q: Hiw, Find radius and intervat fallowing po of Convergence 3. wer serier:

A:

Q: find fix) and

A:

Q: 6x:-ld 212CK) 's aving of integer num 12 and fo), gaJE Z, ) such That 9Cx) = Prove Fhat dey (fagx))…

A: Given  Ring Z12(x)  f(x) = 2+5x+4x4  g(x) = 1+x+x2+3x3    So clearly,  degree of f(x) = 4  degree of…

Q: let it be T=11,2,3 } S:la,b,c] Proved P(TNS)E P(S)NP(T)

A:

Q: & 18) Vaing showj that the idently in

A: Using the identity 1+x2n=1+xn1+xn,                . . . (1) we want to show that 2nn=∑k=0nnk2.…

Q: 17. Use Theorem 4.7 to prove that Vx + 3 2 V3 + x/4 for

A: Let,

Q: 1- 9/if it of Prove that yś(-)- TT

A: In the question it is asked to prove y'(0)=π for the given function y=2xcos-1(2x)-1-4x2.

Q: H PROVE : กะ1 [-¹-², ¹+ ²] = [1,1] 1 t

A: As per our guidelines we are supposed to answer only one question at a time. Kindly repost other…

Q: Is the stries cm vergent or divergent? (Ghow how you kenowi) 2 3n n'+ In + 13

A: a The given series is ∑n=1∞23n.We know that the series ∑n=1∞1n is divergent.And…

Q: prove for n=l,2,3. e that +2+3+..+h = ncn+l) 2. %3D %3D

A:

Q: 1 Use convolution theorem to evaluate L-1 (s²+9)²)

A:

Q: Suppose X~N(112,16), and Z~N(0,1). Then x = 112, is equivalent to z = ______.

A:

Q: Fin Sup(S) and Inf(S), Where 1 S = {1. п

A: Consider the set S=1, 12, 13, ... , 1n, .... Observe that as the value of n approaches to infinity…

Q: (п-1)! d) DE (x"-1nx)

A:

Q: Prove the following: (A → B) ^ (B → A) (D ∨ E) → B A → (~E ∧ C) ———————— ∴~E

A: We prove the given argument by using rules of inference.