Let X = {−1,0,1} and consider the inner product on F(X) defined as follows: ⟨f, g⟩ = f (−1)g(−1) + 2f (0)g(0) + 4f (1)g(1). Let E be the subspace of all even functions defined on X. Set g(x) = x2 + x for every x ∈ X. Find the orthogonal projection of g onto E.

Q: Answer the following: A SIAII OT VOITOUG 4. Determine if the odd functions (f(-x) = -f(x)) are a…

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Q: Let P3 be the vector space of polynomial functions of degree less than or equal to 3 over the field…

A: Let P3 be the vector space of polynomial functions of degree less than or equal to 3 over the field…

Q: Determine if the set W of vectors (x1 , x2) in R such that xr? + x2 = 1 is a subspace of R’.

A: W is not a subspace of R²

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A: We satisfy the clouser and scaler multiplication and show that 0 vector is also in H so H is a…

Q: 6. Let Vi and V2 be subspaces of R" such that Vi C V2. Show that any vector v E R" can be uniquely…

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Q: 3. Determine whether X1 W E R? : x² + x2 = | X2 is a subspace of R².

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Q: Let V be a vector space, T a linear operator on V, and v, w E V. Suppose that T(v + w) = v – 2w and…

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Q: 3. Consider the set C(R) = {continuous functions of real numbers and consider its subsets. Show that…

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Q: For a polynomial p ao + a₁x + a₂x² + + anx, let p denote the polynomial obtained from p by removing…

A: Note:  As you requested only Part C(I) so I answered here only that.

Q: Let P3(R) denote the R-vector space of real polynomials of degree 2 or less p=a+bX+cX2, where a, b,…

A: Given the polynomial .Let the subspace of  be .The subspace is the set of all polynomials from  such…

Q: Define a function f : C -> C by f(x+iy) = (x+2y) + i(3x+4y) for x,y in R. Show that f is additive…

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Q: a Let H = :a – 76 + 3c – d = 0}. First show that H is a subspace of M2×2. Then determine dim(H). d

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Q: Let V be the set of all pairs (x,y) of real numbers together with the following operations: (x1,y1)…

A: Given  be the set of all pairs  of real numbers together with the following operations:

Q: Prove that V is a vector space with the operations

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Q: Let T: P₂ P₂ be a mapping defined by T(a + a₁x + a₂x²) = ao + a₁ (x + 1) + a₂(x + 1)². a) Determine…

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Q: Let P₂ be the vector space of polynomials of degree at most 2. Let T: P₂ → P₂ by T (p(x)) = p(x) + x…

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Q: V = R[x]3, W = R[x]6 be the vector spaces consisting of zero and polynomials in z of degree <3 and…

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Q: 2. Let P3(R) be the vector space of all polynomials of degree 3 or less (and the zero polynomial).…

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Q: Determine if the set W of vectors (X1 , X2) in R such that x1² + x2 = 0 is a subspace of R?.

A: We have to determine whether W is a subspace of R^2 or not.

Q: 13. Prove that if V and W are vector spaces and T: V → W is a linear transformation then Range(T) is…

A: Subspace of a vector space

Q: 1. Determine whether w - {[:]•r:a-4-0} X1 E R? : : x1 – 23 = X2 is a subspace of R?.

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