3. Consider the canonical basis (,,} of R³ and let F = {fi, fa, fs) be defined by (1,0,-1), 72 (0,1,2) and 7=(2, 1, 1). = Find a linear map : R³ R³ such that y()=7 for i=1,2,3.

Q: 2. Let v, = (1,2,1), v2 = (2,9,0), v3 = (3,3,4). Show first that S = {v1, V2, V3} %3D is a basis for…

A: 1. A finite set of vectors v1,v2,...,vn of a vector space V over a field F is said to be linearly…

Q: Consider the inner product space (P2, (, )) with (p, q) = p(to)q(to) + p(t1)q(t1) + p(t2)q(t,),…

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Q: 2. Let x = (1, 1, 1, 1)" and y = (8, 2, 2, 0)". (a) Determine the angle 0 between x and y. (b) Find…

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Q: 7 Given the inner product space (R³(R),+,·,· ), where xoy = 3x₁y₁ + 2(x₂Y₁ + X₁Y₂) + 3x₂Y/₂ + 2x3V3,…

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Q: Let V be the plane with equation z₁ +222 + 3x3=0 in R³. Find the matrix A of the orthogonal…

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Q: Consider the vectors i = (2, 5, 1) and i= (-2, 4, 3). (a) nitude w| = 8? Give a brief explanation of…

A: Given vectors are                                                                u→=2, 5, 1v→=-2, 4,…

Q: 4. Given two bases (b) = transformation y : R² → R² with respect to the basis (f) and let X, be a…

A: Given two bases b=b1,b2 and f=f1,f2 of ℝ2  where  b1=12, b2=34, f1=11, f2=23 Let Af be a matrix of a…

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: (1,1,..., 1)T be the column vector consisting of all 1s. Prove 0. 3. Let a = that La

A: Given: Let a=(1,1,......1)T  Then objective is to prove  La=0

Q: V is the linear span of the polynomials a =x' + 2x – 2x + 1, ß=x' + 3x² - x+4, y = 2x +x - 7x – 7…

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Q: 8. Let V={(x,y)x,y € R}. Define addition on V as follows: (x»x)+(x,y)=(x +2x, 1,y+Y,+2) Find the…

A: To find zero vector of V

Q: 3) Let S contain vectors of the form f(x) = a + bx + cx² + dx³ : f(-2) = 0 and f'(-4)= 0: Then a…

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Q: 2: Show that a map T is linear if and only if T(Au + v) = XT(u)+ T(v) for all A ER and vectors u, v.

A: A linear transformation satisfies  T:U-V  Then T(u+v) = T(u) +T(v)          T(au) = aT(u)  For all a…

Q: Q1. If O (0,0,0) in (x,y,z) 3-D space and r(2,5,1) draw the vector in 3D and find its direction.

A: Plot point in 3D

Q: if u = <1,-3,1> and v = <2,2,2> then find the projection of u onto v

A: The objective of this question is to find the projection of vector u onto vector v. The projection…

Q: Suppose {u, } form an orthogonal basis in R2 and y = 2ū - 7 where u = (8) 6 If W = Span{}, what is…

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Q: Given K= (-2,-1), L=(2,4), M= (-3,-1) and N= (1,4), Prove that KĽ and MN are equivalent by Showing…

A: Two vectors are Equivalent if they have same magnitude and direction.

Q: 5) Consider the vector ū = (1,2, –2) and the points Q(5,5, 2) and R(3,0,0). Find the orthogonal…

A: The orthogonal projections is found below

Q: Consider the inner product on R2 defined by ((x, x2), (y, y2)) = x11 + 4x2y2. Which of the following…

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Q: 7.4Find a subset of the vectors v= (1, -2, 1, -1), v2= (0, 1, 2, –1), vz= (0, 1, 2, –1) and v,= (0,…

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Q: s an inner product space, where (a, 5) = 4 – 5i.

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Q: Find ProjS(u) where S is the space spanned by v1= (1/3)(2,1,2) and v2= (1/3)(-2,2,1). Find the…

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Q: 5. Let V = {(a, b, c, d): a + 3b-2c=0}. Show that (-3, 1,0,0)T., (2,0, 1,0), and (0, 0, 0, 1) form a…

A: The given set is .Now

Q: Parts of this question are independent (a) Suppose īj, ūz ā, b,č 4, det[ū, ī2 č =-1. Find det [v, ūz…

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Q: Determine whether the following vectors in R' are linearly independent lineariy dependent: (i)…

A: As per our guidelines I am supposed to solve only one question. So kindly upload the other questions…

Q: 3c

A: Step 1:a) To evaluate the adjoint (or conjugate transpose) of the linear operator T on the inner…

Q: Find the limit. Enter DNE if the limit does not exist. lim (2,y) + (0,0) x² + 14y2

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Q: Let v1 = (1,2,1), v2 = (2,9,0), v3 = (3,3,4) . Show first that S= {v1, v2, v3} is a basis for ℝ^3.…

A: A linear map is a mapping A to B between two vector spaces that preserves the operations of vector…

Q: (ii) Let W = span{ } v = and The orthogonal projection of v onto W is projw(v) = and the component…

A: Given : v=-43-7    and W=span{ -2-32 , -3-1-2 } Aim: to find (a) projw(v)      ,and   (b) perpw(v)

Q: Let B={ }].[],:)} 3 2 LU} be basis for IR? 2 Find [V]a where 15 6.

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Q: 5. Find the orthogonal projection of the vector b = vectors V₁ = 3 and v₂ = -1- 2 onto the subpace E…

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