Theorem 4.16 (Algebraic properties of orthogonal projections). Let U be a finite-dimensional subs...
Theorem 4.16 (Algebraic properties of orthogonal projections).
Let U be a finite-dimensional subspace of V. PU is a linear map.
Prove that range PU = U, and PUu = u for each u ∈ U.
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Theorem 4.16 (Algebraic properties of orthogonal projections). Let U be a finite-dimensional subs...
Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space ...
Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space and U-V s a subspace, then U㊥ U↓-V, (U 1-U, and Pb is well-defined Inspecting the proofs, convince yourself that all that was needed was for U to be finite- dimensional. (In fact, your book does it this way). Then answer the following questions (a) Let V be an inner product space. Prove that for any u V. if u 0, we have proj, Pspan(v)...
Exercise 11.5.9 Let U and V be finite dimensional spaces over F and let θ : U linear map. v be a ...
Vectors pure and applied
Exercise 11.5.9 Let U and V be finite dimensional spaces over F and let θ : U linear map. v be a (i) Show that o is injective if and only if, given any finite dimensional vector space W map : V W such that over F and given any linear map α : U-+ W, there is a linear (ii) Show that θ is surjective if and only if, given any finite dimensional vector space...
Exercise 12.6.3 Let V and W be finite dimensional vector spaces over F, let U be a subspace of V and let α : V-+ W be a surjective linear map, which of the following statements are true and which...
Exercise 12.6.3 Let V and W be finite dimensional vector spaces over F, let U be a subspace of V and let α : V-+ W be a surjective linear map, which of the following statements are true and which may be false? Give proofs or counterexamples O W such that β(v)-α(v) if v E U, and β(v) (i) There exists a linear map β : V- otherwise (ii) There exists a linear map γ : W-> V such that...
Prob 4· Let V be a finite-dimensional vector space and let U be its proper subspace...
Prob 4· Let V be a finite-dimensional vector space and let U be its proper subspace (i.e., UメV). Prove that there exists ф є V, 0 for all u є U but ф 0. such that p(u)
Vectors pure and applied Exercise 5.7.13 Let U be a finite dimensional vector space over F...
Vectors pure and applied
Exercise 5.7.13 Let U be a finite dimensional vector space over F and let a, B: UU he linear. State and prove necessary andsufficient conditions involving α(U) and β(U) for the existence ofa linear map γ : U-+ U with α γ β. When is γ unique and Explain how this links with the necessary and sufficient condition of Exercise 5.7.1 Generalise the result of this question and its parallel in Exercise 5.7.1 to the case...
(7) Let V be a finite-dimensional vector space over F, and PE C(V) In this question, we will show that P is an orthogonal projection if and only if P2P and PP It may be helpful to recal that P is the...
(7) Let V be a finite-dimensional vector space over F, and PE C(V) In this question, we will show that P is an orthogonal projection if and only if P2P and PP It may be helpful to recal that P is the orthogonal projection onto a subspace U if and only if (1) P is a projection, and (2) ran(P)-U and null(P)U (a) Prove that if P is an orthogonal projection, then P2P and P is self-adjoint Hint: To show...
Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº =...
Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº = {f EV* | f(s) = 0 Vs E S}. (a) Prove that sº is a subspace of V* (S may not be a subspace!) (b) If W is a subspace of V and x € W, prove that there exists an fe Wº with f(x) + 0. (c) If v inV, define û :V* + F by ū(f) = f(u). (This is linear and...
8 Suppose V is finite-dimensional and P E L(V) is such that P2 = P. and...
8 Suppose V is finite-dimensional and P E L(V) is such that P2 = P. and || P v|| = || V || for every v E V. Prove that there exists a subspace U of V such that P = Pu.
10 9. Let U be a finite-dimensional vector space and TE LU). Prove the following statements. (a) (5 pts) Let λ be an eigenvalue T whose geometric multiplicity is m, and algebraic multiplicity is ma....
10 9. Let U be a finite-dimensional vector space and TE LU). Prove the following statements. (a) (5 pts) Let λ be an eigenvalue T whose geometric multiplicity is m, and algebraic multiplicity is ma. Then (b) (5 pts) Let u be a cyclic vector of T of period k 2 2 (such that T*(u) 0 but T-(u) 0). Then are linearly independent.
10 9. Let U be a finite-dimensional vector space and TE LU). Prove the following statements. (a)...
Problem 6. Let V, W, and U be finite-dimensional vector spaces, and let T : V → W and S : W → U be linear transformatio...
Problem 6. Let V, W, and U be finite-dimensional vector spaces, and let T : V → W and S : W → U be linear transformations (a) Prove that if B-(Un . . . , v. . . . ,6) is a basis of V such that Bo-(Un .. . ,%) s a basis of ker(T) then (T(Fk+), , T(n)) is a basis of im(T) (b) Prove that if (w!, . . . ,u-, υ, . . . ,i)...
Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space and U-V s a subspace, then U㊥ U↓-V, (U 1-U, and Pb is well-defined Inspecting the proofs, convince yourself that all that was needed was for U to be finite- dimensional. (In fact, your book does it this way). Then answer the following questions (a) Let V be an inner product space. Prove that for any u V. if u 0, we have proj, Pspan(v)...
Vectors pure and applied
Exercise 11.5.9 Let U and V be finite dimensional spaces over F and let θ : U linear map. v be a (i) Show that o is injective if and only if, given any finite dimensional vector space W map : V W such that over F and given any linear map α : U-+ W, there is a linear (ii) Show that θ is surjective if and only if, given any finite dimensional vector space...
Exercise 12.6.3 Let V and W be finite dimensional vector spaces over F, let U be a subspace of V and let α : V-+ W be a surjective linear map, which of the following statements are true and which may be false? Give proofs or counterexamples O W such that β(v)-α(v) if v E U, and β(v) (i) There exists a linear map β : V- otherwise (ii) There exists a linear map γ : W-> V such that...
Prob 4· Let V be a finite-dimensional vector space and let U be its proper subspace (i.e., UメV). Prove that there exists ф є V, 0 for all u є U but ф 0. such that p(u)
Vectors pure and applied
Exercise 5.7.13 Let U be a finite dimensional vector space over F and let a, B: UU he linear. State and prove necessary andsufficient conditions involving α(U) and β(U) for the existence ofa linear map γ : U-+ U with α γ β. When is γ unique and Explain how this links with the necessary and sufficient condition of Exercise 5.7.1 Generalise the result of this question and its parallel in Exercise 5.7.1 to the case...
(7) Let V be a finite-dimensional vector space over F, and PE C(V) In this question, we will show that P is an orthogonal projection if and only if P2P and PP It may be helpful to recal that P is the orthogonal projection onto a subspace U if and only if (1) P is a projection, and (2) ran(P)-U and null(P)U (a) Prove that if P is an orthogonal projection, then P2P and P is self-adjoint Hint: To show...
Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº = {f EV* | f(s) = 0 Vs E S}. (a) Prove that sº is a subspace of V* (S may not be a subspace!) (b) If W is a subspace of V and x € W, prove that there exists an fe Wº with f(x) + 0. (c) If v inV, define û :V* + F by ū(f) = f(u). (This is linear and...
8 Suppose V is finite-dimensional and P E L(V) is such that P2 = P. and || P v|| = || V || for every v E V. Prove that there exists a subspace U of V such that P = Pu.
10 9. Let U be a finite-dimensional vector space and TE LU). Prove the following statements. (a) (5 pts) Let λ be an eigenvalue T whose geometric multiplicity is m, and algebraic multiplicity is ma. Then (b) (5 pts) Let u be a cyclic vector of T of period k 2 2 (such that T*(u) 0 but T-(u) 0). Then are linearly independent.
10 9. Let U be a finite-dimensional vector space and TE LU). Prove the following statements. (a)...
Problem 6. Let V, W, and U be finite-dimensional vector spaces, and let T : V → W and S : W → U be linear transformations (a) Prove that if B-(Un . . . , v. . . . ,6) is a basis of V such that Bo-(Un .. . ,%) s a basis of ker(T) then (T(Fk+), , T(n)) is a basis of im(T) (b) Prove that if (w!, . . . ,u-, υ, . . . ,i)...
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