21: Let L: M22 → M22be defined by L(A): =E [1 2j [3 4,for AinM22. Consider the ordered basis or M22. Find the representation of L with respect to T and S.

Q: Let L: R → R³ be defined by L (a) Find the representation of L with respect to the nat- ural basis S…

A: Given: L:R3→R3 and is defined by, L100=110L010=201L001=101 It can be further defined as,…

Q: Consider the linear transformation T by taking x E R to T(x Ex1 + x2 and y2 = x1 + 2.x2. Find an…

A: Given, linear transformation T by taking x∈ℝ2 to T(x)=y in ℝ2 ,             where…

Q: Let T : R2[x] → R2[x] be given by T(a+bx+cx2 ) = a + b(x+ 1) + b(x + 1)2 . Let v = x + 1. Find a…

A:

Q: In Exercises 13–16, define T : R² –→ R² by T(x) = Ax. Find a basis B for R? with the property that…

A:

Q: 4 Let A = and define T : R? → R? by T(x) = Ax. Find a -2 -1 basis B such that the matrix [T]s is…

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Q: Let V = P2(R), give V the inner product defined by (f,g) = Lo s(1)g(t) dt, and let W = span {1 – x}.…

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Q: Let T V W be an isomorphism. {T(v₁),..., T(vn)} is a basis for W. Prove that if {v₁,..., Vn} is a…

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Q: За1 Let T : P1 → R? be the isomorphism given by T(a1 x + a2) 4a2 Using the basis {5 – 8, 2x} for P1…

A:

Q: Let T : Ra[z] → Ra[r] have the following images on he standard basis vectors: T(1) = 1+ 4z –…

A: We have to Find Matrix of T with respect to the standard basis.

Q: Find a basis for null(T), where T : R^3 → R^2 is defined by T (x, y, z) = (−7x + 2y − 5z, 4x + 9y +…

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Q: 18. Use the Gram-Schmidt process to transform the given basis (w, w, orthogonal basis where w =…

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Q: 3) Let V have a basis {v1, ..., vn}. Let V* := L(V, F). Prove that the map D : V → V* 0 i+j and…

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Q: Let ẞ= {v1, ••• , vn} be an orthonormal basis for Rn. (a) Prove that, for any x and y in Rn, x · y =…

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Q: Let T : R + R be a linear transformation defined by 21 + 2r3- 12 2i + 12 + 13 T( x2 ) = 23 1) Find a…

A: Given linear transformation Tx1x2x3=2x1+2x3-x2x1+x2+x3

Q: 2. Let B₁ be the basis 2²e², re², e² of V = span(2²e², zeª, z), and B₂ be the basis of that the map…

A:

Q: Let QE M,(R) be given such that the columns of Q form an orthonormal basis for R". Prove that Q'Q =…

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Q: 4. (a) Let L be the line in R' spanned by the vector (1,1, 1, 1). Find a basis for the orthogonal…

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Q: 1. Consider the basis B = {(2, –v2), (2, V2)} for R². Define (ỷ, 2) =y,2 +y22. %3D a) Verify that B…

A: Note:- As per our guidelines, We can answer the first question since the exact one wasn’t specified.…

Q: Let B = {1,x, r²} be the standard ordered basis for P2(R). Let T be the linear operator on P2(IR)…

A: We will be using Eigenvectors to solve the problem.

Q: (11) Find the dual basis {V4.V } of the stan dard basis of IR. Let T:IR → IR defined by T(xiy, z)…

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Q: 1. Consider the vector space V = M3x3(C) ôver the neld all matrices of the form 211 Z33 Find a basis…

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Q: Let 3 the the standard ordered basis on M2×2(R), let Y = {1 – x, 1+ x}, and let T: M2x2 (IR) → P1…

A: a = 10 b = 2  The detailed solution is as follows below:

Q: If T: V → W is a linear transformation of finite dimensional vector spaces and Rank(T) = 3, then the…

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Q: . Consider the following linear transformation; T(*1, 22, 23, T4) = (x1 – x2 – 13 – 24, 0, 3r1 - 3r2…

A:

Q: snip

A: (a).               Orthonormalize the set of the vectors                      v1 = 1111 , v2 =…

Q: (a) Prove that the set ẞ = {p0(x), p1 (x), ... , Pn(x)} of Lagrange polynomials is linearly…

A: Given: ẞ = {p0(x), p1 (x), ... , Pn(x)} To Prove" (a). Prove that the set ẞ of Lagrange polynomials…

Q: Given T : R2 --> R2 defined by T (x, y) = (x - 2y,  x + y) (a) Is T is injecting? (b) Determine…

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Q: 15. Use the Gram-Schmidt process to transform the basis {r1, x2, a3, T4} into an orthogonal basis…

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Q: Let L : V → W be a linear transformation, and let dim V = dim W . Prove that L is invertible if and…

A: Given: L:V→W is a linear operator and let dim V = dim W

Q: 2.19 Consider an endomorphism : R³ → R³ whose transformation matrix (with respect to the standard…

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Q: 2. Let L: R3 → R³ and let 1 1 [L] = 3 -2 1 0 3 be the matrix of L with respect to the standard basis…

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Q: Find the orthogonal complement W⊥ of W and give a basis for W ⊥

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Q: Consider P2, endowed with the following inner product: 1 1 (p, q) : G P(0)q(0) + G P(1)q(1) + for…

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Q: onsi with its standard product. nat A E Mnxn(R) is orthogonal if and only if, for any orthonormal…

A: since {v1,...,vn} is orthonormal , thenvit * vj=0 if i different than j and 1 if i=j where I wrote…

Q: Let, T: R²³ R³T (x₂√, z) = (x+√+2, 27¹², 24+32) • Test whether the transformation linear or not T is…

A: Given- T: R3→R3: T x,y,z=x+y+z , 2y+z, 2y+3z (a) To test T is linear transformation (b) To find…

Q: 1.Assume (f,g) = | f(x)g(x)dx is the inner product for V = P, . Apply the Gram-Schmidt process to…

A: Orthonormal basis

Q: 1. Prove that if {v1, v2, · · · , vn} is a basis for V and w1, wg, ., w, are vectors in W, not…

A:

Q: Let {u₁, U2, u2} be an orthonormal basis for an inner product space V. Suppose v = au₁ + bu₂ + cuz…

A: We have v=au1+bu2 +cu3 Now, ∥v∥=a2+b2+c2 Given ∥v∥=80 So,  80=a2+b2+c2⇒802=a2+b2+c2⇒80=a2+b2+c2…

Q: Q: Find a representation of Jp(t)> at a time t +0 using the ordered basis {|e1>, ]e2>, ]e3> }.

A:

Q: 6. Define fe (R2)* by f(x, y) 2x + y and T: R2 (3x + 2y, x). (a) Compute T'(f). (b) Compute [T']p,…

A: We will use the basic knowledge of linear transformations to answer all the three given subparts of…

Q: 7.3.12 ND phere y1 natrir representation of T is diagonal, and find that matrix representation.…

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Q: 1.2 Find a basis B for the domain of I Suh that the mahix for ī alative to Bis iagend. Hogand. T:…

A: We have to Find a basis B for the domain of T such that the matrix for T relative to B is diagonal.

Q: (: :) - {(: :)-(: :)-(: :)- C )} be the standard basis for the Let T : P2 → M2x2(R) be the linear…

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Q: Let B the the standard ordered basis on M2x2 (R), let y = {1, x}, and let T : M2x2 (R) → P1 (R) be…

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Q: Let T: P2 that is, 7. > P2 be the linear operator T(p(x)) = p(2x+ 1), T(co+ C,x+ C,x²) = c, + c,(2x…

A:

Q: 5. a) Compute the change of basis matrix from 8 = ( ) to Ez in R?. 7() - A b) Given an explicit…

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Q: →>> Determine the criterion of the transformation T, where T: R² R² is defined on the basis B =…

A:

Q: Let B={v_1...v_n} be a basis for R^n and let E be a standard basis for R^n. Let T: R^n ->R^n be…

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Q: d) Consider the linear map T: C→ C whose matrix in the standard basis e1, e2, e3 is (0 1 0V 0 0 1 0…

A: Given linear map T:ℂ3→ℂ3 whose matrix in the standard basis e1,e2,e3 is 010001000. Then, the linear…

Contingency Table

A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.

Binomial Distribution

Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.