Linear map Definition and 59 Threads

Homework Statement Find the linear map f:R^2 \rightarrow R^3, with f(1,2) = (2,1,0) and f(2,1)=(0,1,2) Homework Equations The Attempt at a Solution I actually don't understand this task. PLease help! Thank you...

[SOLVED] form of a linear map Homework Statement Say E is a linear space (not necessarily of finite dimension), and R is the real numbers. Say we have a (contiuous) linear form T from E x R to R. Can we say T is of such and such a form? Particularily, can we say that T=g1+g2 where g1:E-->R...

hello, I've been reading some proofs and in keep finding this same argument tyo prove that a linear map is injective viz, we suppose that t(a,c) = 0 and then we deduce that a,c = 0,0. is it the case that the only way a linear map could be non injective is if it took two elements to zero? i.e. t...

Homework Statement Give a specific example of an operator T on R^4 such that, 1. dim(nullT) = dim(rangeT) and 2. dim(the intersection of nullT and rangeT) = 1 The attempt at a solution I know that dim(R^4) = dim(nullT) + dim(rangeT) = 4, so dim(nullT) = dim(rangeT) = 2. I also...

Homework Statement This is a problem related to linear map over vector spaces of functions and finding kernels. Let V be the vector space of functions which have derivatives of all orders, and let D:V->V be the derivative. Problem1: What is the kernal of D? Problem2: Let L=D-I,where I...

I need help. For this problem, you have to use the Gram-Schmidt process to make it orthogonal. My trouble is finding the bais for the kernel of the linear map L: R4 -> R1 defined by L([a,b,c,d)]=a-b-2c+d I know the dimension of the kernel is 3, but how? I have tried setting it...

Homework Statement Consider the linear map A : R3 ----> R3 given by A(x1, x2, x3) = (x1 − x2,−x1 + x2, x3). (a) Find the adjoint map A^*. (b) Obtain the matrix representations of A and A* with respect to the canonical basis f_1 = [1, 2, 1], f_2 = [1, 3, 2], f_3 = [0, 1, 2]...

I'm trying to understand what one-forms are. The book I'm reading says a one-form is a linear map from a vector to a real number. It uses the gradient as an example but isn't the gradient a map from a function to a vector?

What are the relations between a matrix H and its transpose H^T? I am not asking about the relations between the coefficients, I am asking the relations as linear maps (H: F^m->F^n; H^T: F^n->F^m). I am not sure exactly how I should pose the question actually, but I am thinking there is some...