One to one function(Injective): A function is called one to one if for all elements a and b in A, if f(a) = f(b),then it must be the case that a = b.Equality: Two functions are equal only when they have same domain, same co-domain and same mapping elements from domain to co-domain.Addition and multiplication: let f1 and f2 are two functions from A to B, then f1 + f2 and f1.f2 are defined as-:.Image and Pre-Image – b is the image of a and a is the pre-image of b if f(a) = b.Range – Range of f is the set of all images of elements of A.Domain and co-domain – if f is a function from set A to set B, then A is called Domain and B is called co-domain.means f is a function from A to B, it is written as If b is the unique element of B assigned by the function f to the element a of A, it is written as f(a) = b. A is called Domain of f and B is called co-domain of f. Mathematics | Rings, Integral domains and FieldsĪ function f from A to B is an assignment of exactly one element of B to each element of A (A and B are non-empty sets).Mathematics | Independent Sets, Covering and Matching.
ONTO VS ONE TO ONE EXAMPLES DISCRETE MATH SERIES
Mathematics | Sequence, Series and Summations.Mathematics | Generating Functions – Set 2.Discrete Maths | Generating Functions-Introduction and Prerequisites.Mathematics | Total number of possible functions.Mathematics | Classes (Injective, surjective, Bijective) of Functions.Number of possible Equivalence Relations on a finite set.Mathematics | Closure of Relations and Equivalence Relations.Mathematics | Representations of Matrices and Graphs in Relations.Discrete Mathematics | Representing Relations.Mathematics | Introduction and types of Relations.Mathematics | Partial Orders and Lattices.Mathematics | Power Set and its Properties.Inclusion-Exclusion and its various Applications.Mathematics | Set Operations (Set theory).Mathematics | Introduction of Set theory.ISRO CS Syllabus for Scientist/Engineer Exam.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
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