Let V be a real vector space. Recall the following definition Definition 1. An operator L...
3.2.1: Let V be a vector space with Basis B and let L be an operator on V. L2 means the operator applied twiceL2(v) = L(L(v)). Show that the Matrix of L2 is the square of the matrix of L, i.e LL? Demonstrate this problem for the space V span (1, t, t2, t3) and let L d/dt (so L2d/dt) SO
Let V be a finite-dimensional vector space and let T L(V) be an operator. In this problem you show that there is a nonzero polynomial such that p(T) = 0. (a) What is 0 in this context? A polynomial? A linear map? An element of V? (b) Define by . Prove that is a linear map. (c) Prove that if where V is infinite-dimensional and W is finite-dimensional, then S cannot be injective. (d) Use the preceding parts to prove...
6. Let L be the linear operator mapping R3 into R3 defined by L(x) Ax, where A=12 0-2 and let 0 0 Find the transition matrix V corresponding to a change of basis from i,V2. vs) to e,e,es(standard basis for R3), and use it to determine the matrix B representing L with respect to (vi, V2. V
Let A be an invertible linear operator on a finite-dimensional complex vector space V. Recall that we have shown in class that in this case, there exists a unique unitary operator U such that A=UA. The point of this exercise is to prove the following result: an invertible operator A is normal if and only if U|A= |AU. a) Show that if UA = |A|U, then AA* = A*A. Now, we want to show the other direction, i.e. if AA*...
1. let V be a vector space and T an operator on V (i.e., a
linear map T: V--> V). Suppose that T^2 - 5T +6I = 0, where I is
the identity operator and 0 stands for the zero operator
...
Read Section 3.E and 3.F V) 1. Let V be a vector space and T an operator on V (i.e., a linear map T: V -» Suppose that T2 - 5T + 61 = 0, where I is...
Let V = Cº(R) be the vector space of infinitely differentiable real valued functions on the real line. Let D: V → V be the differentiation operator, i.e. D(f(x)) = f'(x). Let Eq:V → V be the operator defined by Ea(f(x)) = eax f(x), where a is a real number. a) Show that E, is invertible with inverse E-a: b) Show that (D – a)E, = E,D and deduce that for n a positive integer, (D – a)" = E,D"...
2. Let V be the vector space of polynomials in two variables r and y of degree at most two: V-(ar' + bry + суг + dr + ey + f | a,b,c, d, e, f E R} Let T be the linear operator on V defined by Find the Jordan canonical form of T
2. Let V be the vector space of polynomials in two variables r and y of degree at most two: V-(ar' + bry + суг...
11. =(7.5), #,(-3,-1) 2) Let = (1.-5). v. =(-2,2) and let L be a linear operator on R whose matrix representation with respect to the ordered basis . is a) Determine the transition matrix (change of basis matrix) from, v,to (1) (Draw the commutative triangle). 3 b) Find the matrix representation B, of L with respect to ,v} by USING the similarity relation
Let (V,〈 , 〉v) and (W.〈 , 〉w) be finite-dimensional inner product spaces. Recall that the adjoint L* : W → V of a linear function L Hom(V,W) is completely determined by the equation <L(v), w/w,-(v, L* (w)של for every v є V and w є W . Use this to prove the following facts: (a) (Li + L2)* = Lİ + L: for Li, L26 Horn(V,W) (b) (α L)* =aL' for a R and L€ Horn(V,W) (c) (L*)* =...
#3 Only
In the following 4, let V be a vector space, and assume B- [bi,..., bn^ is a basis for V. These 4 problems, taken together, give a complete argument that the coordinate mapping Фв : V → Rn defined by sending a vector v E V to its coordinate vector [v]в є Rn is an isomorphism between V and Rn. In other words, Фв : V-> Rn is a well- defined linear transformation that is one-to-one and onto....