Theorem 5.30. Let T be a linear operator on a finite-dimensional vec- tor space V, and let W₁, W2, ..., W be T-invariant subspaces of V such that V=W₁ W₂W. For each i, let ß, be a basis for W₁ and ß = B₁ B₂Bk. If A = [T]p and A₁ = [Tw], for i = 1, 2,..., k, then A= A, Θ Α Θ …ΘΑ.. 2 Proof. Exercise.

Q: 1. Let M22 be the vector space of all real 2 × 2 matrices. Consider W= D = 2} be a subset of M22.…

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Q: 1. Let B = be a 1 basis for the subspace V. Find the matrix that projects any 4 vector v E R* onto…

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Q: 4. Decide if the following statement is True or False: Let U, W, and X be subspaces of a finite…

A: Given - Let U, W and X be subspace of a finite dimensional vector space V such that U ∩ W = 0 and U…

Q: 11. Find a basis of the following subspace of R4. W = all vectors of the form (a, b, c, d) where a +…

A: To find: Basis of W={(a,b,c,d) | a+b-c+d=0}.

Q: Which of the following sets of vectors is a basis for the subspace W of R' given by W = ¿x = z + :…

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Q: 4. Assume that T E L(V) and that U1 and U2 are T-invariant subspaces of V. Prove that U1 n U2 is…

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Q: a. Find a nonzero element X in W. 16 X = Let V be the vector space of symmetric 2 x 2 matrices and W…

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Q: Let T be defined by T(x) = Ax. Find a vector x whose image under T is b, and determine %3D whether x…

A: Let us consider the given system of equation AX=b then the system has  If number of equation is more…

Q: Find a basis and the dimension of each of the following subspaces of R" и a. W1 = v: u+ v = 0 and u…

A: For (a), Now, we need to find the basis. Here, u+v=0⇒v=-u and u-2w=0⇒w=u2. So,…

Q: a. Find a nonzero element X in W. X Let V be the vector space of symmetric 2 x 2 matrices and W be…

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Q: 6. The set B = {v = (1,1,1),u = (0,0,–3), w = (2,2,0)} %3D A linearly dependent in R³. B Linearly…

A: B={v=(1, 1, 1), u=(0, 0, -3) and w=(2, 2, 0)} To check if Bis linearly dependent or linearly…

Q: Suppose S = {r, u, d} is a set of linearly independent vectors. If x = 2r+u+ 5d, determine whether T…

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Q: a. Find a nonzero element X in W. X = Let V be the vector space of symmetric 2 x 2 matrices and W be…

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Q: ent, find a linear relation. a) vi = (2, 1, 0), V2 = (0, 1,0), V3 = (-1,2,0) %3D b) wi = (1,2, –2),…

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Q: 0-0- Let ui = u2 = and set W = {w e Rª : w · v1 = w · v2 = 0}. W is a subspace of R4. Find a basis…

A: Find the dimension first and find vectors which satisfies given two relation

Q: Let V be a vector space over R. Show that if T is a linear operator on V, the following conditions…

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Q: Let {u,., up} be an orthogonal basis for a subspace W of R" and {vr, ., vg} be an orthogonal basis…

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Q: Use the graph attached. The linear function is f(x). The quadratic function is g(x). Use the…

A: Let us assume two functions f(x) and g(x) such that f(x) is a linear function with a negative…

Q: of R4. Create a basis for U. Consider the subspace X = U = = span{ 2 B 1 08- ‚x}

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Q: 9. Let F be a field. Find bases of the following two subspaces of F5 : W₁ = {(a₁, a2, a3, a4, a5) =…

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Q: Let V be a finite-dimensional R-vector space. Let 3 and 3' be two ordered bases for V, and let Q =…

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Q: If T is defined by T(x) = Ax, find a vector x whose image under T is b, and determine whether x is…

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Q: Let D = (d₁.d₂.d3) and F= {f₁.f2.13) be bases for a vector space V, and suppose f₁ = 6d, -d₂ +4d3,…

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Q: 33. Let V - P3 be the vector space of polynomials of degree <3, and let p₁(x) = (2-x)² and p₂(x) =…

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Q: 4. Let H be the set of 2 × 2 whose trace is equal to zero. That is [a11 a12 Н- E M2(R) | a11 + a22 =…

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Q: Let Uj and U2 be vector subspaces of a vector space V over the real numbers. Suppose that V1, V2, V3…

A: We shall answer first question only as you’ve asked more than one question. For others kindly post…

Q: 11. Let V be an inner product space and let B1, ..., B, be any independent vectors in V. The may…

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Q: The vector x is in a subspace H with a basis ß= (b₁, b2]. Find the B-coordinate vector of x. -6-0-3…

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Q: Let T and T, be linear operators on V and let W be a subspace of V. Show that if W is invariant…

A: Let T1 and T2 be linear operators on V and let W be a subspace of V. If W is invariant under both…

Q: Suppose S = {r, u, d} is a set of linearly independent vectors. If x = 5r+2u + 4d, determine whether…

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Q: If W, and W2 are finite - dimensional subspaces of a vector space V, then W + W2 is finite –…

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Q: Suppose S={r,u,d} is a set of linearly independent vectors. If x=3r+5u+3d, determine whether…

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Q: Let T be a linear operator on the space V, and let W, ,W hecessary and sufficient condition that…

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Q: [W1 W2 E R3: w1 – 2w2 + 3w3 = 0} is a subspace of W3 %3D 1. Show that the set W = R3.

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Q: If M is linear subspace of hilbert space H then M is clos

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