Example 3. The operator Ax = d/dt x(t), defined on the subspace M of C[0, 1] consisting of all continuously differentiable functions, has range C[0, 1]. A is not bounded, however, since elements of arbitrarily small norm can produce elements of large norm when differentiated. On the other hand, if A is regarded as having domain D [0, 1] and range C[0, 1], it is bounded with || A|| = 1.

Q: If W is a subspace in an inner product space V and V = W + W, prove that V = WW¹.

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Q: *1. Which of the following are subspaces? Justify your answer in each case. (d) (x € R³ : x² + x² +…

A: The given sets are  (d) x∈ℝ3:x12+x22+x32=1 (f) x∈ℝ3:x12+x22+x32=-1 (h) x∈ℝ3:x=301+s211+t121 for some…

Q: Suppose Vis finite-dimensional, T = L(V), and U is a subspace of V. Prove that U and U are both…

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Q: 7. Let V be a complex vector space and let T e L(V). Let r be the minimal polynomial of T and…

A: For a complex vector space V any subset of V suppose Q will be it's subspace if Q itself is a vector…

Q: Consider the subspace spanned by B = {cos(x), sin(x), x cos(x), x sin(x)} inside the space of…

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Q: Use the inner product = = f(x)g(x)da in the vector space Cº [0, 1] to find the orthogonal projection…

A: Given: The functions are fx=2x2+4, gx=x-12 and hx=1. The inner product is given by f,g=∫01fxgxdx in…

Q: Let V be a vector space, and let W₁,. Let W = .... W₁ + + Wk = {w₁ + + wkw₁ € W₁, ..., wk € Wk}. (a)…

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Q: A function f: R → R is called even if f(-x) = f(x) for all x ER, and a function f: R → R is called…

A: Given that Ue denotes the set of all real valued even functions and Uo denotes the set of all real…

Q: Let U be a subspace of the vector space P100 (R), with dim(U) = 37. Suppose W is another subspace of…

A: P100(R) is a vector space  of all polynomial of degree 100 and smaller. The dimension is 101.For…

Q: 6. Let space V be the space of all polynomials of degree 3: V = {f(x) = ax³ + ba² + cx + d}. Space W…

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Q: M2x2 (R) is a vector space over R. Make use of this result to show that the subset S defined by a S…

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Q: Find the standard matrix for the orthogonal projection of R3 onto the subspace spanned by aj=(1,1)…

A: The vectors that span the subspace are a1=1, 1 and a2=0, 1. The normalising vectors are 12, 12 and…

Q: Suppose W₁, W₂ are subspaces of a finite-dimensional inner product space. Prove the following: (a)…

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Q: 5. Let x, y be vectors in R4, and let L be the set L = {ax +By: a, ß ER, a + B = 1}. (a) Verify that…

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Q: 5. Let V be the vector space of all continuous functions on [1,5]; that is, V = C[1,5]. -5 Let W be…

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Q: Determine whether the given set S is a subspace of the vector space V. A. V is the vector space of…

A: First noted that if  be a vector space over a field  be a subspace of ,then null vector  of V must…

Q: 6. Let C"(R) denote the sct of all real-valued functions defined on the real line that have a…

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Q: 6. Let space V be the space of all polynomials of degree 3: V = {f(x) = ax + ba² + cx + d}. %3D…

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Q: The subset {p(x)|p(1) = 2p(0)} in the space P of all polynomials.

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Q: 3: Determine whether the following subsets are subspaces a) H = { ||zy = 0} CR". b) H = { x+ 2y + 3z…

A: We have to find

Q: 4. Consider the vector space C[-1, 1] of continuous functions on [-1, 1] with inner product (f,g) =…

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Q: Problem 5. Determine whether the subspace of R(-1,1) consisting of all differentiable func- tions f:…

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Q: 4. Let S be the subspace of P₂ defined by S = span{1 − x, 1 + x, 5, 1 – x², 2x}. Find dim(S), and…

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Q: 9. Determine if the following subsets of R² are subspaces of R²: (a) S = {(x,y)| xy ≥ 0} (b) E=…

A: 9.  (a) The given set is S={(x,y) | xy≥0 } . Let (1,2)∈Sand (-2,-1)∈S( Since 1×2=2>0 and…

Q: Suppose T E L(V) and U is a subspace of V. (a) Prove that if U null T, then U is invariant under T.…

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Q: 3. Let A be an arbitrary m x n matrix. Consider all x's, such that Ax = 0, and show that the set of…

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Q: Which of the following subsets is not a vector subspace in the vector space R2[r]?

A: The correct option is:e){ a+bx∣a+b−1=0}Explanation:

Q: (b) Let X C[0, 1], be the the space of continuous functions on the interval [0, 1]. %3D Consider X…

A: The given problem is related to a normed space of continuous functions on an interval. Given:…

Q: 9. Determine if the following subsets of R² are subspaces of R²: (a) S = {(x,y)| ry >0} (b) E =…

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Q: Consider Mnxn (R) the set of n x n matrices with real entries. Define T: Mnxn (R)→ Mnxn (R) by…

A: Given:Mn×nR is the set of all n×n matrices with real entries.T : Mn×nR →Mn×nR is defined…

Q: What is the dimension of the sum of subspaces V and W in R3[x], where V={f∈R3[x]∣ f(0)=0} and…

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Q: Suppose V is finite-dimensional and dim V > 1. Prove that the set of noninvertible operators on V is…

A: Given that V is a finite dimensional vector space with dimV>1.We need to prove that the set of…

Q: Let V, W be a pair of subspaces of Rn and suppose that W ⊂ V . Prove step by step in detail that…

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Q: 6) Suppose U = {(x1,x2, ...) E F®|x; # 0 for only finitely many j}. (a) Show that U is a subspace of…

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Q: Show that the set of all functions./ E C(R) such that df / dt exists anddf/ dt = 0 is a…

A: Let W=f∈CR/f'=0. Let f,g∈W and α∈R. Then  αf+g=αf+gx=αfx+gx=αfx+gx⇒αf+g'=αf'+g'=0+0, since f,g∈W=0…