Let y be any element in the co-domain (N), such that f(x) = y for some element x in N (domain).Īlso f is not a bijection. X = y (We do not get ± because x and y are in N that is natural numbers) Let x and y be any two elements in the domain (N), such that f(x) = f(y). Now we have to check for the given function is injection, surjection and bijection condition. Classify the following function as injection, surjection or bijection: (i) f: N → N given by f(x) = x² Which of the following functions from A to B are one-one and onto? (i) f1 = So, both are not same. Let y be any element in the co-domain (Z), such that f(x) = y for some element x in Z So, different elements of domain f may give the same image.
ISRO CS Original Papers and Official Keys.GATE CS Original Papers and Official Keys.This says that, for instance, R 2 is “too small” to admit an onto linear transformation to R 3. Each row and each column can only contain one pivot, so in order for A to have a pivot in every row, it must have at least as many columns as rows: m ≤ n. The matrix associated to T has n columns and m rows. If T : R n → R m is an onto matrix transformation, what can we say about the relative sizes of n and m ? Tall matrices do not have onto transformations Of course, to check whether a given vector b is in the range of T, you have to solve the matrix equation Ax = b to see whether it is consistent. To find a vector not in the range of T, choose a random nonzero vector b in R m you have to be extremely unlucky to choose a vector that is in the range of T. Whatever the case, the range of T is very small compared to the codomain. This means that range ( T )= Col ( A ) is a subspace of R m of dimension less than m : perhaps it is a line in the plane, or a line in 3-space, or a plane in 3-space, etc. Suppose that T ( x )= Ax is a matrix transformation that is not onto. The previous two examples illustrate the following observation. Note that there exist tall matrices that are not one-to-one: for example,Įxample (A matrix transformation that is not onto) This says that, for instance, R 3 is “too big” to admit a one-to-one linear transformation into R 2.
Each row and each column can only contain one pivot, so in order for A to have a pivot in every column, it must have at least as many rows as columns: n ≤ m. If T : R n → R m is a one-to-one matrix transformation, what can we say about the relative sizes of n and m ?
Wide matrices do not have one-to-one transformations If you compute a nonzero vector v in the null space (by row reducing and finding the parametric form of the solution set of Ax = 0, for instance), then v and 0 both have the same output: T ( v )= Av = 0 = T ( 0 ). All of the vectors in the null space are solutions to T ( x )= 0. This means that the null space of A is not the zero space. By the theorem, there is a nontrivial solution of Ax = 0. Suppose that T ( x )= Ax is a matrix transformation that is not one-to-one. The previous three examples can be summarized as follows. Hints and Solutions to Selected ExercisesĮxample (A matrix transformation that is not one-to-one).3 Linear Transformations and Matrix Algebra