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1. Suppose that N is finite and suppose that we have a probability mass function f...
(e) Let GLmn(R) be the set of all m x n matrices with entries in R...
(e) Let GLmn(R) be the set of all m x n matrices with entries in R and hom(V, W) be the set of all lnear transformations from the finite dimensional vector space V (dim V n and basis B) to the finite dimensional vector space W (dimW m and basis C) (i) Show with the usual addition and scalar multiplication of matrices, GLmRis a finite dimensional vector space, and dim GCmn(R) m Provide a basis B for (ii) Let VW...
12. (True/False) (a) Let AE Rm*n . Then R(A) (b) Let AERm*n. Then N(A) is isomorphic to N(AT) (c) We define < A. B > = Tr (BTA ) where A, B E Rnxn . is isomorphic to R(A Then 〈 . , . 〉 is an...
12. (True/False) (a) Let AE Rm*n . Then R(A) (b) Let AERm*n. Then N(A) is isomorphic to N(AT) (c) We define < A. B > = Tr (BTA ) where A, B E Rnxn . is isomorphic to R(A Then 〈 . , . 〉 is an inner product on Rmxn. (d) Consider a periodic-function space V with period of 1 sec. Define an inner product on V by <f,a>= f(t )a (t ) dt. Then cos 2 π t...
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for...
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for all x € R2,AG R, but is not a linear transformation. (b) Show that a linear transformation VWfrom a one dimensional vector space V is com- pletely determined by a scalar A (e) Let V-UUbe a direet sum of the vector subspaces U and Ug and, U2 be two linear transformations. Show that V → W defined by f(m + u2)-f1(ul) + f2(u2) is...
Problem 5. Given a vector space V, a bilinear form on V is a function f : V x V -->R satisfying the following four c...
Problem 5. Given a vector space V, a bilinear form on V is a function f : V x V -->R satisfying the following four conditions: f(u, wf(ū, ) + f(7,i) for every u, õ, wE V. f(u,ū+ i) = f(u, u) + f(ū, w) for every ā, v, w E V. f(ku, kf (ū, v) for every ū, uE V and for every k E R f(u, ku) = kf(u, u) for every u,uE V and for every k...
Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear f...
Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear form on V. (b) Let B = (1, ... ,T,) be a basis of V. Prove that there exists a unique inner product on V making Borthonormal. (c) Let (V) be the set of all inner products on V. By part (a), J(V) C B(V). Is J(V) a vector subspace of B(V)?...
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)...
For each n E N, define a function fn A - R. Suppose that each function fn is uniformly continuous. Moreover, suppose there is a function f : A R such that for all є 0, there exists a N, and for all x...
For each n E N, define a function fn A - R. Suppose that each function fn is uniformly continuous. Moreover, suppose there is a function f : A R such that for all є 0, there exists a N, and for all x E A, we have lÍs(x)-f(x)|く for all n > N. Then f is uniformly continuous. Note: We could say that the "sequence of functions" f "converges to the function" f. These are not defined terms for...
1/2 b dr Problem 1: Suppose that [a, b] exists R, and let V be the space of all functions for which and is finite. For...
1/2 b dr Problem 1: Suppose that [a, b] exists R, and let V be the space of all functions for which and is finite. For two functions f and g in V and a scalar A e R, define addition and scalar multiplication the usual way: (Af)(x) f(x) f(x)g(r) and (fg)(x) Verify that the set V equipped with the above operations is a vector space. This space is called L2[a, b
1/2 b dr Problem 1: Suppose that [a,...
In this problem we consider only functions defined on the real numbers R. A function f is close to a function g if 3x E R s.t. Vy E R, A function f visits a function g when Vz E R, R s.t. a<y and...
In this problem we consider only functions defined on the real numbers R. A function f is close to a function g if 3x E R s.t. Vy E R, A function f visits a function g when Vz E R, R s.t. a<y and f() -g) For a given function f and n E N, let us denote by n the following function: n(x)-f(x)+2" Below are three claims. Which ones are true and which ones are false? If a...
1. (25 pts) Let f(x) be a continuous function and suppose we are already given the Matlab function "f.", with header "function y fx)", that returns values of f(x) Given the following...
1. (25 pts) Let f(x) be a continuous function and suppose we are already given the Matlab function "f.", with header "function y fx)", that returns values of f(x) Given the following header for a Matlab function: function [pN] falseposition(c,d,N) complete the function so that it outputs the approximation pN, of the method of false position, using initial guesses po c,pd. You may assume c<d and f(x) has different signs at c and d, however, make sure your program uses...
(e) Let GLmn(R) be the set of all m x n matrices with entries in R and hom(V, W) be the set of all lnear transformations from the finite dimensional vector space V (dim V n and basis B) to the finite dimensional vector space W (dimW m and basis C) (i) Show with the usual addition and scalar multiplication of matrices, GLmRis a finite dimensional vector space, and dim GCmn(R) m Provide a basis B for (ii) Let VW...
12. (True/False) (a) Let AE Rm*n . Then R(A) (b) Let AERm*n. Then N(A) is isomorphic to N(AT) (c) We define < A. B > = Tr (BTA ) where A, B E Rnxn . is isomorphic to R(A Then 〈 . , . 〉 is an inner product on Rmxn. (d) Consider a periodic-function space V with period of 1 sec. Define an inner product on V by <f,a>= f(t )a (t ) dt. Then cos 2 π t...
Problem 3 (LrTrmations). (a) Give an example of a fuction R R such that: f(Ax)-Af(x), for all x € R2,AG R, but is not a linear transformation. (b) Show that a linear transformation VWfrom a one dimensional vector space V is com- pletely determined by a scalar A (e) Let V-UUbe a direet sum of the vector subspaces U and Ug and, U2 be two linear transformations. Show that V → W defined by f(m + u2)-f1(ul) + f2(u2) is...
Problem 5. Given a vector space V, a bilinear form on V is a function f : V x V -->R satisfying the following four conditions: f(u, wf(ū, ) + f(7,i) for every u, õ, wE V. f(u,ū+ i) = f(u, u) + f(ū, w) for every ā, v, w E V. f(ku, kf (ū, v) for every ū, uE V and for every k E R f(u, ku) = kf(u, u) for every u,uE V and for every k...
Problem 6. Let V be a vector space (a) Let (--) : V x V --> R be an inner product. Prove that (-, -) is a bilinear form on V. (b) Let B = (1, ... ,T,) be a basis of V. Prove that there exists a unique inner product on V making Borthonormal. (c) Let (V) be the set of all inner products on V. By part (a), J(V) C B(V). Is J(V) a vector subspace of B(V)?...
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)...
For each n E N, define a function fn A - R. Suppose that each function fn is uniformly continuous. Moreover, suppose there is a function f : A R such that for all є 0, there exists a N, and for all x E A, we have lÍs(x)-f(x)|く for all n > N. Then f is uniformly continuous. Note: We could say that the "sequence of functions" f "converges to the function" f. These are not defined terms for...
1/2 b dr Problem 1: Suppose that [a, b] exists R, and let V be the space of all functions for which and is finite. For two functions f and g in V and a scalar A e R, define addition and scalar multiplication the usual way: (Af)(x) f(x) f(x)g(r) and (fg)(x) Verify that the set V equipped with the above operations is a vector space. This space is called L2[a, b
1/2 b dr Problem 1: Suppose that [a,...
In this problem we consider only functions defined on the real numbers R. A function f is close to a function g if 3x E R s.t. Vy E R, A function f visits a function g when Vz E R, R s.t. a<y and f() -g) For a given function f and n E N, let us denote by n the following function: n(x)-f(x)+2" Below are three claims. Which ones are true and which ones are false? If a...
1. (25 pts) Let f(x) be a continuous function and suppose we are already given the Matlab function "f.", with header "function y fx)", that returns values of f(x) Given the following header for a Matlab function: function [pN] falseposition(c,d,N) complete the function so that it outputs the approximation pN, of the method of false position, using initial guesses po c,pd. You may assume c<d and f(x) has different signs at c and d, however, make sure your program uses...
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