1) Let X be a complex vector space and f2: X R be a linear functional. Define f:X C by f(x) = f2(ix) + if2(x) Show that f is linear.

Q: Please solve this question

A: Note: Properties of trace: Let A and B be two matrices. trace(A+B) = trace(A) + trace(B). trace(αA)…

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Q: , Let C" (R) be the vector space of "smooth" functions, i.e., real-valued functions f(x) in the…

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Q: suppose Vn is the vector space of polynomials of degree <n-1 and with real cofficients. Determine…

A: As per the question we are given a vector space Vn of polynomials with degree ≤ n-1 and with real…

Q: Let A = → Rm be (a;j)ij be an m xn matrix of real numbers, and let f : R" defined by x2 X2 f = A…

A: I am solving this in step 2

Q: (6) Suppose that T is a linear operator on a 2 dimensional vector space V and that T+ a I for any a…

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Q: if T is an operator in a real vectorial space with characteristic polynomial (x²-2x+5)². What are…

A: For the solution follow the next steps.

Q: Theorem 7. Let T be a linear transformnation from space U (F) into a vector space V (F). and only…

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Q: Determine the kernel and the range of the linear operator on L(x) = (x1 + x3, X2, 0)

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A: Given statement is that if Q:ℝn→ℝ is a quadratic form, thenQ(x) attains its maximum and minimum…

Q: b) g(x) = S,* In(2r + 1)dr

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Q: Determine a scalar r such that Ax = rx, where 1 A =| 1 1 and x = 4 -4 5

A: Ax=r x ⇒ x is an eigenvector of Afor the eigenvalue r. So to find r, we compute all the eigenvalues…

Q: Prove that the function : R-{1} →R- {2} defined by 2a f(x)= s bijective. 2 1

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Q: Let H1 and H2 be any two Hilbert Spaces and T : H1 → H2 be a bounded linear operator. Let M1 = N(T)…

A: Given:- Let H1 and H2 be any two Hilbert Spaces and T : H1 → H2 be a boundedlinear operator. Let M1…

Q: Conclude that there is a vital assumption when using rank nullity, often overlooked. What is it? Let…

A: According to Rank-nullity theorem, if V and W are vector spaces and T is a linear operator from V…

Q: let T be a linear operator on a finite- dimensional vector space V, and let f(t) be the…

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Q: Let L: R2 → R2² be the linear operator defined by L 2

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Q: Determine a scalar r such that Ax = rx, where

A: Matrix: A=12-11014-45x=-12141AX=12-11014-45.-12141

Q: 3. Let C[0, 1] be equipped with the inner product (f, g) = f f(x) g(x) dx, where f = f(x) and g =…

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Q: Verify that the set C[a, b] of all continuous real-valued functions defined on the interval a ≤ x ≤…

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Q: The function = ] f(x) g(x) dx defined for vector space C[-1, 1] satisfies : Select one: a.…

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Q: Exercise 4.12 Give an cxample where f(x) - b as x → a and g(y) + c as y – b but g(f(x)) + c as x → a

A: To find an example where fx→b as x→a and gy→c as y→b but gfx→c as x→a. Let b∈ℝ be a fixed real…

Q: If f(x) = x^2 + 7x+2, g(x) = x^2 - x + 1 find a polynomial h(x) of degree 2 such that together these…

A: Given:  f(x)=x2+7x+2,g(x)=x2-x+1 We have to find a polynomial hx of degree two such that together,…

Q: Determine a scalar s such that A2 x = sx when Ax = r x.

A: Given that Ax=rx. Multiply both sides of the above equation by A and simplify as follows.

Q: If T: V → V is a linear operator with minimal polynomial х — 2, then T is diagonalizable. True False

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Q: Let P2 be the vector space of all polynomials of degree ≤ 2 with coefficients in R, and S= {1 + 2x,…

A: The basis for the vector space P2 is 1, x, x2. We have to write the matrix form for the set S. We…

Q: Let Te B(H) be an invertible operator. Show that a(T-) = 0(T)-.

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Q: et L:RR be a linear operator such that L x2 )-X1, then Im(L) spanned by= X3 X1 Gelect one: a. O b.…

A: T : A  → B thenIm(T) = { TX : X ∈ A}

Q: 1.7) Example: Let P (R) the vector space of real functions , define by: = f(x)g(x) dx Is inner…

A: Given: Pn(ℝ) the vector space of real functions, define f, g by: f, g=∫01f(x)g(x)dx Is inner…

Q: If f(x) and g(x) have a period p, show that h(x)3Da f(x) + b g(x) (a, b constants) has the period p.…

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Q: Let Pn denote the vector space of polynomials in the variable x of degree n or less with real…

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Q: b.) answer. Let f = x² and g = (1-x)². Are x and y orthogonal in this inner product space? Justify…

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Q: 2) Find the indicated norms of the following linear functionals defined on RS b) f(x1,X2, X3, X4,X5)…

A: This question is about application of functional analysis

Q: Let f1(x) = 3x and f2(x) = ∣x∣. Graph both functions on the interval −2 ≤ x ≤ 2. Show that these…

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Q: /- X be the vector Space of all Complex 2x2 and de fine T: x →x by Tx = bx, where Let matrices bEX…

A: Given :-  Let X be the vector space of all complex 2×2 matrices and define T : X →X  by Tx = bx ,…

Q: Theorem 7. Let T be a linear transformnation from space U (F) into a vector space V (F). Then T is…

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Q: 6. Find a basis for the null space of the functional f defined on R³ by ƒ(x) = §1 +§2 − §3, where x…

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Q: Show that the functions f (x) = |x + 1|and g(x) = x² are linearly %3D independent on the interval I…

A: Given that f(x)=|x+1| So, f(x)=x+1 in [1, ∞) g(x)=x2

Q: Verify that the set C[a, b] of all continuous real-valued functions defined on the interval a ≤ x ≤…

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Q: let T be a linear operator on a finite- dimensional vector space V, and let f(t) be the…

A: Given that T be a linear operator on a finite-dimension vector space V, and f(t) be the…

Q: Let P, denote the vector space of polynomials in the variable x of degree n or less with real…

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Q: Let V be a finite dimensional vector spac there exists a linear functional f € V* such f(v) * 0.

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Q: Let T be a linear operator on the finite – dimensional vector space

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Q: Let R2[x]be the vector space of real polynomials of degree < 2. Let f: R2x] - R2x]be the…

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Q: Suppose L: R → R° be defined by u, +17 Lu2 2u2 %3D Uz Uz Determine whether L is a linear operator or…

A: According to question given :L:R3→R3Lu1u2u3=u1+12u2u3We have to determine whether L is a linear…

Q: Determine a scalar r such that Ax = rx, where

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.