9. Let T be an operator on an n-dimensional vector space V over a field F. (a) If T has n distinct eigenvalues prove that T is diagonalizable. If T is diagonaziable prove that the characteristic polynomial of T splits over F. (c) Give an example of an operator T such that the characteristic polynomial of T splits but T is not diagonalizable.

Q: (a) Let Pn(R) denote the vector space of polynomials of degree at most n, where addition is the…

A: Given :  T : P3(ℝ) → P2(ℝ)  defined by                  T(p(x)) = 3p''(x)-2p'(x) To find :(a) We…

Q: 6. Suppose V is a complex vector space with dimension n and T E L(V) is invertible. Let p and q be…

A: In the question it is asked to prove p(0)q(z)=znp(1z).

Q: 4x-y<-1

A: Given  4x-y&lt;1 Can be rewritten as y&gt;4x-1 So we take the area above the line y = 4x -1 And all…

Q: B = (v₁, V₂, (The ia Let V be an inner product space (over C or R), with an orthonormal basis Vn)…

A:

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

A:

Q: (b) Let v, 2,... , vn be a basis of a vector space V. Suppose f: V -> V is a lincar map so that Vi…

A: To work out the required properties of the linear operator f.

Q: Let V=R2[x] be the real vactor space of polynomials of degree most 2. B={f0 ,f1 ,f2 } where f0(x)=1,…

A: The quadratic approximation of a function f at x0 is: f(x0)+f'(x0)(x-x0)+f''(x0)(x-x0)22 = Q(x)

Q: 5. Consider the space P2 (the space of polynomials of degree at most 2) and define the inner product…

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Q: Find a basis for the T-cyclic subspace W of the operator T: V→V generated by the vector 2: V=R³,…

A: Given vector space is V=ℝ3 and the operator T:V→V is defined as: Ta,b,c=a+b+c,3b,a-c. We need to…

Q: Let V = Poly4(R) and let B = (1, x2 − 1, x, x^3 − 2x, x^4 − 3x^2 + 1). (a) Find a basis C of V such…

A: Let V = Poly4(R) and let B = {1, x2 − 1, x, x3 − 2x, x4 − 3x2 + 1}. Note that here V is a vector…

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

A:

Q: 9. Find an orthonormal basis for P[r]3 with a basis 1, 2, 2² if the inner product on P[r]3 is…

A: We have to find the orthonormal basis for  with a basis The inner product is defined by

Q: 3. Let TV → V be an operator on a finite-dimensional space V. (a) Prove that T is not invertible if…

A:

Q: In the vector space of real functions on [-, π], let the inner product be defined by ㅠ (f,g) = f( […

A: Using the Gram-Schmidt orthogonalisation prosess to find an orthogonal basis.

Q: Let V be a finite-dimensional inner product space. A linear operator U on V is called a partial…

A: A partial isometry is a linear operator on a finite-dimensional inner product space that preserves…

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

A:

Q: 2. Linear Maps (a) Let Pn denote the space of polynomial of degree at most n with real coefficients.…

A: We have to find the matrix representative of differential operators using standard monomial basis.

Q: 6. (a) Let D : P3 P3 be the differentiation operator. Find the kernel and range of D. (b) Let M2.2…

A: We know that if T : U→V be a linear transformation from the vector space U to the vector space V,…

Q: 1 Let P₂[x] be the set of all polynomials of order at most 2. Show that {1, (x - 1), (x - 1)(x - 2)}…

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Q: 5) Prove directly that two finite dimensional vector spaces over the same field are iso- morphic iff…

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Q: Let A be a real symmetric matrix. Show that all eigenvalues of A are nonneg- ative if and only if A…

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Q: (4). Let x = (a). Determine an elementary reflector R such that Rx lies on the x-axis. (b). Extend x…

A: 4) Solution:Given that, Now find the following:

Q: Let V be the vector space of polynomials of degree < 2 with real coefficients, endowed with the…

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Q: (a) Show that the list {p o(x), p 1(x), p 2(x), ...,p n(x),...,} where po(x) = 1, p1(x) = 1+x, P;(x)…

A:

Q: W= {x: where x are eigenvectors of an A } Is W a subspace of R" ? Show the prove

A: See solution below

Q: (a) Let f: V > V be any linear map where V is a vector space of dimension n over a field K. Suppose…

A: Let f : V --&gt;V be any linear map where V is a vector space of dimension n over a field K. Suppose…