3.5.4 ([Ber14, Ex. 3.6.14]). Let T : V → W and S : W → U...

3.5.4 ([Ber14, Ex. 3.6.14]). Let T : V → W and S : W → U be linear maps, with V finite dimensional.

(a) If S is injective, then Ker ST = Ker T and rank(ST) = rank(T).

(b) If T is surjective, then Im ST = Im S and null(ST) − null(S) = dim V − dim W

Solutions

Expert Solution

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

1. Let U = {r, s, t, u, v, w, x, y, z}, D = {s,...

1. Let U = {r, s, t, u, v, w, x, y, z}, D = {s, t, u, v, w}, E = {v, w, x}, and F = {t, u}. Use roster notation to list the elements of D ∩ E. a. {v, w} b. {r, s, t, u, v, w, x, y, z} c. {s, t, u} d. {s, t, u, v, w, x, y, z} 2. Let U = {r, s, t, u, v, w, x, y, z},...

Let V be a vector space and let U and W be subspaces of V ....

Let V be a vector space and let U and W be subspaces of V . Show that the sum U + W = {u + w : u ∈ U and w ∈ W} is a subspace of V .

Questionnnnnnn a. Let V and W be vector spaces and T : V → W a...

Questionnnnnnn a. Let V and W be vector spaces and T : V → W a linear transformation. If {T(v1), . . . T(vn)} is linearly independent in W, show that {v1, . . . vn} is linearly independent in V . b. Define similar matrices c Let A1, A2 and A3 be n × n matrices. Show that if A1 is similar to A2 and A2 is similar to A3, then A1 is similar to A3. d. Show that...

Let V and W be Banach spaces and suppose T : V → W is a...

Let V and W be Banach spaces and suppose T : V → W is a linear map. Suppose that for every f ∈ W∗ the corresponding linear map f ◦ T on V is in V ∗ . Prove that T is bounded.

If V = U ⊕ U⟂ and V = W ⊕ W⟂, and if S1: U...

If V = U ⊕ U⟂ and V = W ⊕ W⟂, and if S1: U → W and S2: U⟂ → W⟂ are isometries, then the linear operator defined for u1 ∈ U and u2 ∈ U⟂ by the formula S(u1 + u2) = S1u1 + S2u2 is a well-defined linear isometry. Prove this.

Let W denote the set of English words. For u, v ∈ W, declare u ∼...

Let W denote the set of English words. For u, v ∈ W, declare u ∼ v provided that u, v have the same length and u, v have the same first letter and u, v have the same last letter. a) Prove that ∼ is an equivalence relation. b) List all elements of the equivalence class [a] c) List all elements of [ox] d) List all elements of [are] e) List all elements of [five]. Can you find more...

Let u and v be orthogonal vectors in R3 and let w = 3u + 6v....

Let u and v be orthogonal vectors in R3 and let w = 3u + 6v. Suppose that ||u|| = 5 and ||v|| = 4. Find the cosine of the angle between w and v.

Let A ∈ L(U, V ) and B ∈ L(V, W). Assume that V is finite-dimensional....

Let A ∈ L(U, V ) and B ∈ L(V, W). Assume that V is finite-dimensional. Let X be a 3 × 5 matrix and Y be a 5 × 2 matrix. What are the largest and smallest possible ranks of X, Y, and XY? Give examples of the matrix to support your answers

Calculus dictates that (∂U/∂V) T,Ni = T(∂S/∂V)T,Ni – p = T(∂p/∂T)V,Ni – p (a) Calculate (∂U/∂V)...

Calculus dictates that (∂U/∂V) T,Ni = T(∂S/∂V)T,Ni – p = T(∂p/∂T)V,Ni – p (a) Calculate (∂U/∂V) T,N for an ideal gas [ for which p = nRT/V ] (b) Calculate (∂U/∂V) T,N for a van der Waals gas [ for which p = nRT/(V–nb) – a (n/V)2 ] (c) Give a physical explanation for the difference between the two. (Note: Since the mole number n is just the particle number N divided by Avogadro’s number, holding one constant is equivalent...

True/False Question: If sppan{u,v}=W where u not equal to v. Then dim(W)=2. Answer: False Reasoning let...

True/False Question: If sppan{u,v}=W where u not equal to v. Then dim(W)=2. Answer: False Reasoning let u=1, v=2 then span(1,2}=R but dim(R)=1 not 2. I know the answer is false. please tell me whether my reasoning is correct.

Subjects