If T : R6 [a] → M2x2 is onto then Nullity(T) = 3. O True O False

Q: Let z= y- 4a, a u'v and y-v- 3u. Compute On kum)-(1,1) -

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Q: 69. Let F(x, y) = (x,0). Prove that if C is any path from (a, b) to (c, d), then F. dr (o2

A: Given: F(x, y) = <x, 0>. C is any path from (a, b) to (c, d).

Q: Prove that if w=f(x, y, z), x=r-s, y=s-t, and z=t-r then dw dw dw + + = 0. dr ds ôt

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Q: . Finish showing the Complementary Slackness Theorem by establishing that under the hypothesis of…

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Q: Prove that r(z) = (In-)²-'dt , Re(z) > 0.

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Q: by f(x) = (x2 + 1)35 for all x e Ris A. One-One but not onto B. onto but not one-one C. neither…

A: Given f(x) = (x2 + 1)35

Q: Define T R → R by T (a, b, c) = (a – c, b+ c) Is T one to one? [ Type Y for yes and N for no. Is T…

A: T  can't be one-one because of domain and codomain but T is onto

Q: Given that w = f (u, v) And u = xz. , V = dw = X az aw aw Prove that Z - ду

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Q: (4b) Is Tâ onto? How do we know?

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Q: Ti Ik'→ IR? be a linear transtormation 3). let defined by -E: Co all ER T C3 x2 + Show that T is…

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Q: If Ea, is divergent and Eb, is divergent, then E(a, +bn) is divergent. Select one: O True O False

A: Sum of two divergent series not necessarily divergent.

Q: Exercise 4 Show that for cach k EN it holds that kn-k < n! S n", Vn E N, n2 k. Deduce from this that…

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Q: Prove that T2 is chaotic on 1.

A: Given that T2=2x0&lt;x&lt;122(1-x)12&lt;x&lt;1 The maximum for T is 1 at x=0.5. The dynamics of T2…

Q: 2. Prove that if AB = AC, then B need not equal C unless A is invertible.

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Q: Consider the D-operator P(D) = mD² + 2kD +¢, where m and c are positive. If -c/m < k< 0, then

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Q: If S and T are normal linear operators satisfying ST* = T* S and TS* = S*T, show that their sum S+T…

A: I have used just definition of normal operator

Q: If the i, j entry of A is i times j, show that det A = 0. (Exception when A = [ 1 ] .)

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Q: The Wronskian of y,(t)=e 2, y,()=e" is-e St. Select one: True False

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Q: Prove that for a E N, a always invertible in Z,2+3a+1·

A: We know that an element a∈ℤn is called invertible if there is an element b∈ℤn such that a+b≡0  mod…

Q: Give a natural deduction proof of V x B(x) from hypotheses 1. Vx (A(x) V B(x)) and 2. V y (- A(y))

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Q: Let T : V --&gt; W be linear, b ϵ W, and K = {x ϵ V: T (x) = b} be nonempty. Prove that if s ϵ K,…

A: Given: Let T : V --&gt; W be linear, b ϵ W, and K = {x ϵ V: T (x) = b} be nonempty The objective is…

Q: if x, y>o such that x Sy then Vã svy Prove by direct method

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Q: @ Let A -a Find Cenditiens on a,b.c and d that AX =o has infailely many Solution. what must there…

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Q: Q1:- Let T =- where W-N(0, 1) and V~xShow that T2-F(1,r).

A: We know that  The ratio of chi-square divided by its degrees of freedom is called the…

Q: Prove that there do not exist a, b ∈ Z satisfying 54a − 39b = 2.

A: Solution:

Q: In R* whire n eN, let こ and イ.ES":(tx € E),(ヨr >o)ud tlnt Br (x) CE lang the deginition 4 topelogs…

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Q: 1 if i= j 0 otherwise. P:(c;) = 5 is a linearly independent set.

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Q: Let x, y, A, u E Z. Prove that if 1 = Ax + µy then r, y are relatively prime.

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Q: Let -{[:);}--{:)[:} -{{)}} Both B and C are bases for R2. If {x}s =: -[:], what is {x}c?

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Q: Define T:R → R' by T(x, y, z) = (x-5y+4z, y- 6z). a) Is T one-to-one? Justify your answer. b) Is T…

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Q: Let f(r, y) = In (z³y) (1,3) = 3. then Select one: O True O False

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Q: 2. Use Wronskian to show that fi = x, f2 - Inz, and fa C (R). z Inz are linearly independent in

A: First we will calculate the wronskian of given functions f_1,f_2,f_3

Q: 3. Let T: M₂2 → R² be defined as T ( [])=(a - b, cd). Determine whether or not I' is a linear…

A: I have given the answer in the next step. Hope you understand that

Q: . Suppose that T e L(V) and U and W are invariant under T. Prove that U + W is also invariant under…

A: If U is subspace of V and T(U) is subeset of U then U is invariant under T

Q: Let f(x, y) = In(x²y), where a = s² + t² and y = 2s + 3t. %3D Which of the following is a correct…

A: Given,           f(x,y) = lnx2y3,  where x = s2 + t2 and y = 2s + 3t

Q: Let S-(xeZ vmod12-8) and T (xe2| 4x). Prove that SCT but TgS La s-(rez| 3reZz, x-12r+7) and T-(xeZ|…

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Q: Please solve...

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Q: Show that |Re(z)|+|Im(z)| < v2|z| for any z e C.

A: The solution is given as

Q: Suppose A and B are invertible then det(A) = det(B) %3D O True O False

A: Consider A and B are invertible

Q: at x and y are orthogonal

A: Given: x and y in ℝn, x+y=x-y To show:  x and y are orthogonal in ℝn if and only if x + y = x − y

Q: Prove (x + y) + z = (x + z) + y without using induction.

A: It can be solve using group theory

Q: -If CI(A) denotes the closure of A, which of the following is not correct in general: O (AU B)' = A'…

A: option (1) is not correct see below the reason

Q: Suppose that T :C C is R-linear. Show that T is C-linear if and if T(iz) = iTz for all z.

A: First we describe the definition of C-linear function:

Q: 2) Suppose S,T E L(V) are such that ST = TS. Prove that null S is invariant under T.

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Q: Let x(v, w) = } In(v² + w²). Show that = 0

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Q: Let fons =loag, Cln). Find t'ot at

A: Given, f(x)=logx(ln(x))

Q: Prove that (aA + bB)t = aAt + bBt for any A , B E Mmxn(F) and any a,bϵ F.

A: Consider, aA+bBt=aAt+bBt    ∵A+Bt=At+Bt=aAt+bBt     ∵kAt=kAt for any scalar k.=aAt+bBt

Q: (a) Verify the Cayley-Hamilton theorem. (b)  Use the theorem to express A2 and A3 as linear…

A: Part (a), Let, A=4-112 Now, the characteristic equation of A,…

Q: Let f(x, y, z) = x2y + y2z. Use the Chain Rule to calculate af /as and af /at (in terms of sand t),…

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Q: 2. Use Wronskian to show that fi = x, f2 = ln x, and f3 C (R). x In x are linearly independent in

A: ANSWER: