a b Consider M₂ (R) {[%d) | a, b, c, d = R с X Y (a) Show that I = { [ + ² ] | x₁yER} is a subring of M₂ (R). Y X Z Z (b) Is J = ZER an ideal of I? Justify your answer. Z Z = }, a ring under matrix addition and matrix mutiplication.

Q: 2. Which of the following is a basis for the space of polynomials of degree < 2 over R? (Check all…

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Q: he following function: =: Find fxy. f(x, y) = 5x4y6x4y5 - 21

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Q: Without using anything from the above two results show that for any complex inner- product (·, ·)v…

A: We need to show that, x,yV=xU, yUH In other words for any finite dimensional inner-product space,…

Q: 25. B = (f1, f2, f3) is a basis for R2[x] where fi = 1+ x + 2x², f2 = 1 + 2x + 5x², f3 = 2 + x +…

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Q: If L and U are invertible, then (LU) – Find A from the given LU factorization: 9 -27 7-14 96 = -63…

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Q: Consider P2 (R) together with inner product: Find an orthogonal basis for P2 (R). (p(z),9(2) = Г'р…

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Q: 2. Let (X,(x) and (Y, (,)) be Hilbert spaces. For (x1,y1), (2, y2) = X x Y, put ((x1,y1), (2, 2))xxy…

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Q: Consider a square matrix A having characteristic polynomial p(A) = (1 + 1) ^ 2 (A - 2)(^ – 3) ^ 2 a)…

A: Dimension of a matrix : Let A be a square matrix with n rows &amp; n columns .So its dimension is…

Q: Let s = {1, 2, 3} andT: Fun(S) + R' be the transformation T) = (10), s) – 2/), 1() – 12) and…

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Q: Find (u, v), ||u|l, ||v||, and d(u, v) for the given inner product defined on R". u = (1, 1, 1), v =…

A: Our guidelines we are supposed to answer only three subpart. Kindly repost other subpart as the next…

Q: 1. Prove that if u and v are vectors in Rn, then |u·v| s ||u|||v|| where ju v| denotes the absolute…

A: As per guidelines we have to attempt only 1st question

Q: True or False. The set {(0,0),(1,0),(2,0),(3,0),(4,0)} is a principal ideal of z5 x z3

A: Ideal:  A non - empty set S of ring R  is said to be an ideal of ring R if, (i)    a - b  ∈ S ;  ∀…

Q: 10. (a) Define: Basis 6 -1 -2 2 3 -4 5 8 (c) Find dim(nul(B)) and dim(col(B)), does your result…

A: The given matrix B=-36-11-11-223-12-458-4. We have find a basis for the null space of B and the…

Q: Find (u, v), ||u||, ||v||, and d(u, v) for the given inner product defined on R". u = (5, 0), v =…

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Q: Use the functions f and g in C[-1, 1] to find (f, g), ||F||, ||9||, and d(f, g) for the inner…

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Q: Let L: R2 R2 be defined by L 4*]-[***] represents the coordinates of v in the standard ordered…

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Q: Let Consider the complex inner product space C³ with the usual inner product (u, v) = ₁V₁ + U₂ V₂ +…

A: Inner products space

Q: Ax = b must have a solution for eac In Exercises 33 and 34, T is a linear transformation from R2…

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Q: Take L: V --> W, from P2 --> P2 and L(p) = p + p', with basis B = {1 + x + x2, 1 + x, 2x + x2} and…

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Q: (c) 0 is neither diagonalizable nor

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Q: (a) R=Z, I = 35Z.

A: Note: We’ll answer the first question since the exact one wasn’t specified. Please submit a new…

Q: Let V=R₂₂ [x] with (the inner product, 1 <f(x), g(x)) = f(x) g(x) dx Find a basis for the or…

A: Answer is not correct, since the second element is not orthogonal to x2+x+1. Please check and verify…

Q: Let B = {B₁ = [], B₂ = 1²²= 1·²=6 D be a basis of M2x2 (R). Suppose that T: M₂×2 (R) → P₂(R) is a…

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Q: Recall that the Euclidean scalar product in R² is defined by (x1, Y1) · (x2, Y2) := X1T2+ Y1Y2 - (i)…

A: Since you have posted multiple questions but according to guidelines, we will solve first question…

Q: 7. Let V={(x,y) x,y € R}. Suppose addition and scalar multiplication are defined using the following…

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Q: e) Let W = {(x; y) x,yER}. Define addition and scalar multiplication %3D a + B = (x; y) + (a ; b) =…

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Q: Suppose that c is the line segment from (a, b) to (c,d). (a) Show that [rd |x dy – y dx = ad – bc…

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