Prove that rotations, reflections, and composites of rotations and reflections are orthogonal operators.

Q: a + ht+A2h?t2 + A3h³t3 + • .. = f(a)+ f'(a)(t + A2t + A3t° + · .·)+ ½f"(a)(t + A2t (2.138) + Azt…

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Q: find a counterex for property 7 gives matrices A e Mzxz and BEM3x3 and scalar c So that C (A #B) #…

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Q: Give an example of an orthogonal operator that is neither a reflection nor a rotation.

A: Consider a linear operator T defined as:

Q: Prove that the adjoint operator is linear and bounded

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Q: Explain why S is not a basis for R². S = {(-6,4)} OS is linearly dependent. OS does not span R². O S…

A: Given: S=-6, 4. We have to explain why S is not a basis for R2.

Q: Show that the set S= {(-2,-4), (1, 2)} is the basis of R2 or not?

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Q: Determine whether the set of vectors {(2,-1,3),(4,1,2),(8,-1,8)} form a basis for R3.

A: Consider the vectors : 2,-1,3, 4,1,2, 8,-1,8 To form a basis these vectors, the vectors must be…

Q: Prove that (a) the zero transformation and (b) the identity transformation are linear…

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Q: Explain why S is not a basis for P2. S = {1 - x, 1 - x², -1 - 4x + 5x2} O sis linearly dependent. O…

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Q: Show that B = {v1=(1,0,0), v2=(3,7,-2), v3=(0,4,1)} is a basis for R3

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Q: Check whether the set V= {(2, 3), (3, 4)} is basis of R2 or  not?

A: this a spanning set and linearly independent so the vectors forms a basis of R^2

Q: Represent the vector ⟨16,8,−14⟩ in terms of the ordered basis B={⟨5,−2,−2⟩,⟨1,−4,2⟩,⟨−1,−5,3⟩}.

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Q: Define Vector addition.

A: For the addition of two vectors, we can use the parallelogram law.  The Sum of two sides equals the…

Q: 2) Let V = P5(Q) the vector space consisting of polynomials in the variable x of degree at most 5.…

A: Firstly we define linear transformation, then we show given transformation is a linear and then we…

Q: Determine the bilinear transformation that maps the points (1– 2i,2 + i,2 + 3i) into the points (2+…

A: (1-2i , 2+i , 2+3i) and (2+i , 1+3i , 4)

Q: Determine whether the resulting linear transformations are linearly dependent. Provide necessary…

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Q: 3. Let V = P3(Q) be a vector space of polynomials with coefficients in Q and variable x of degree at…

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Q: Determine whether (V₁, V2, V3) is a basis for R³. V₁ = No O Yes V₂ = 8 s 3 V3 -2 [-7]

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Q: T: P2 P2, .2 T(ax2 + bx + c) = ax² + bx +c B = {1, 1 – x, (1 - x)²} B' = {1, x , x²} v = x² – 3x +3…

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Q: 3. Let B = Consider T : R2,2 → R²,² defined by T(A) = AB – BA. Let [o o] Bị and B2 :

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Q: If two Hermitian operators A and B commute, show that they can be simultaneously diagonalized.

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