All four questions solved
prove properties 2,4,6,8 of theorem 1.1 3. Theorem 1.1 R” as defined above has the following...
Let be an arbitrary mapping satisfying the properties (S1) - (S4) of Theorem (at the end). Beyond that let Show that the following statements apply to all u, v ∈ Rn. The Theorem: For the scalar product, vectors u, v, w∈ Rn : We were unable to transcribe this imageWe were unable to transcribe this imageu_u We were unable to transcribe this image u_u
1. Prove the following properties of scalar multiplication and addition for vectors. Let s,t e R and v,w E Rn (a) (st)v s(tv) (b) (st)(vw) = sv + tv + sw + tw) (c) Use a special form of w and part (b) to instantly prove (s + t)v = sv + tv. 1. Prove the following properties of scalar multiplication and addition for vectors. Let s,t e R and v,w E Rn (a) (st)v s(tv) (b) (st)(vw) = sv...
4. Prove the following statement: Consider the ODE x = f(x) with x : J C R → Rn and f : Rn → Rn. If a continuously differentiable real-valued function V = V(x) exists such that (a) V is defined on Bs(0) {x E Rn : Irl < δ} (b) V(x) 0 for x E Bs(0) 1 fo) (c) V 0) 1 (o then the origin is unstable. (x) >0 for rE Bs 4. Prove the following statement: Consider...
(6) (This question does not relate to the above conditions.) Prove that the following system of trigonometric functions is an orthonormal system of L?(-7,7): cos no, sin ne 27 n=1,2,.. Moreover, set f(0) = 62. Write the Fourier expansion off with respect to the system of trigonometric functions in L'(-, 7). Problem 2. We define k00 Example. Let N be a null set. If u(x) = v(x) for x® N, then u(x) = v(x) a.e. Similarly, if lim uk(x) =...
Problem statement: Prove the following: Theorem: Let n, r, s be positive integers, and let v1, . . . , vr E Rn and wi, . . . , w, є Rn. If wi є span {v1, . . . , vr} for each i = 1, . . . , s, then spanfVi, . .., v-) -spanfvi, . .., Vr, W,...,w,) Suggestiorn: To see how the proof should go, first try the case s - 1, r 2..] Problem...
Prove Theorem 4.2.21. The Singular Value Decomposition. PROVE THAT IF MATRIX A element of R^n*n Theorem 4.2.21. Let A e Rnxn. Then ||A| Definition 4.2.2. On R" we will use the standard inner product (7.7) = .2.2015 j=1 | 7 ||2=1 Theorem 4.2.20. Let A € R"X". Then ||A||2 = 01. Proof: Let AE Rnxn and let Let A=USVT be an SVD of A. We have || A||2 = max || 17 || 2 = max, ||UEV17 || 2 =...
Let U be an open subset of R. Let f: U C Rn → Rm. (a) Prove that f is continuously differentiable if and only if for each a є U, for each E > 0, there exists δ > 0 such that for each x E U, if IIx-all < δ, then llDf(x)-Df(a) ll < ε. (b) Let m n. Prove that if f is continuously differentiable, a E U, and Df (a) is invertible, then there exists δ...
Implicit Function Theorem in Two Variables: Let g: R2 → R be a smooth function. Set {(z, y) E R2 | g(z, y) = 0} S Suppose g(a, b)-0 so that (a, b) E S and dg(a, b)メO. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above. 2) Since dg(a,b) 0, argue that it suffices to...
Please answer this question Implicit Function Theorem in Two Variables: Let g: R2 - R be a smooth function. Set Suppose g(a, b)-0 so that (a, b) є S and dg(a, b) 0. Then there exists an open neighborhood of (a, b) say V such that SnV is the image of a smooth parameterized curve. (1) Verify the implicit function theorem using the two examples above (2) Since dg(a, b)メ0, argue that it suffices to assume a,b)メ0. (3) Prove the...
7. Consider the Theorem: Suppose A and B are two lower triangular matrices (Defined in 8 3.1), of order n. Then, the product AB is also a lower triangular matrix. Likewise for upper triangular matrices. (We say that the set of lower triangular matrices, of order n, is closed under multiplication.) Prove this theorem, for n = 3, by multiplying the following two matri- ces: a1 0 0 A bi b 0 1 0 0 and B 2 0 21...