Theorem 7.15. (Cayley-Hamilton) For every finite-dimensional vector space over a fieldK, for every linear map f: EE, for every basis (e₁,...,en), if A is the matriz over fover the basis (e₁,...,en) and ifPA(X) = X + C₁X"¹+...+₂is the characteristic polynomial of A, thenPA(f) = f +c₁f¹ + ... + cid = 0.

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Answered: K, for every linear map f: EE, for…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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3. Let T: V→ V be a linear operator on a finite-dimensional vector space V. Let B = {₁,..., Un} be an ordered basis for V. Let k be an integer with 0
a ) Decide whether the mappings given above are linear- Explain properly: if you saythat the mapping is linear, you need to prove that it is linear, if you say that the mapping isnot linear you need to demonstrate why.b)For the mappings given above that are linear find kernel, nullity, rank and set ofimages. Find also basis for the kernel and for the set of images.
Suppose S = {V₁, V2,..., Vn} is linear independent in V. Prove that T= {V₁, V₁ + V2, V₁ + V2 + V3,..., V₁ +...+ Vn} is linearly independent in V. 2023/01/23: This question has been updated so that the indices match.
Suppose S = {r, u, d) is a set of linearly independent vectors. If z = 2r + 4u + 2d, determine whether T = {r, u, z} is a linearly independent set. Select an Answer 1. Is T linearly independent or dependent? If T is dependent, enter a non-trivial linear relation below. Otherwise, enter 0's for the coefficients. r+ u+ z = 0.
7. Let V and W be finite-dimensional vector spaces and T: VW is an isomorphism. Prove that if 3 = {₁,.., Un} is a basis for V, then y = (T(₁) T(vn)} is a basis for W. 8. Let T R² R3 be a linear transformation such th that T(1,2) T(-1,3)= (1,4,2). Find T(x, y) and 7-1(x, y) if possible. = (0,3,-1) and
Proposition 1: Let T: V-> W be a linear transformation of linear spaces V and W, suppose that a set CW is linearly of vectors {v} →→ V1, V2, V3 }} CV has the property that T {T (v₁),T (√₂²),T (√₂)} (v independent. Then, the set of vectors {0₁, 02, 07 } Below is a "proof" of the above proposition. You may use it as a template and upload your proof below. → → → Proof: Suppose that a₁v₁ + a2v₂ + a3v3 Then, applying T to both sides we get Since T is a linear transformation, we get T Since {7 (5),(6). (6) {T T T = V3 CV is linearly independent. Ō. are linear independent we get that
Consider R{(x0, 1,...): x₁ = R} the vector space of countably infinite sequences of reals. We define the following maps: L(xo, 1, 2,...) = (x₁,x₂,...) R(x0, x1,x2,...) = (0, 0, 1, 2, ...) we call L the left shift transformation and we call R the right shift transformation. 1. Prove that L is onto but not one-to-one. 2. Prove that R is one-to-one but not onto. 3. Why are these transformations special? Could such maps exist for a finite dimensional space?

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K, for every linear map f: EE, for every basis (e₁,...,en), if A is over the basis (e. and if