[PDF] [PDF] Linear Algebra Note that a square matrix

Is a matrix injective or surjective?

For square matrices, you have both properties at once (or neither). If it has full rank, the matrix is injective and surjective (and thus bijective)....If the matrix has full rank (rankA=min{m,n}), A is: injective if m?n=rankA, in that case dimkerA=0; surjective if n?m=rankA; bijective if m=n=rankA. What makes a matrix surjective?

What is an injective and surjective function?

A bijection is a function that is both injective and surjective. This means that every element of the codomain appears exactly once. What is the difference between an injective function and a surjective function? An injective function is a function where every element of the codomain appears at most once.

How do you know if a matrix is a surjection?

For example, if T is given by T(x) = Ax for some matrix A, T is a surjection if and only if the rank of A equals the dimension of the codomain. (Recall that the rank of A gives the dimension of the column space of A .) To test injectivity, one simply needs to see if the dimension of the kernel is 0.

Is T surjective or injective?

Hence, T is not surjective since there are vectors in Q2 (e.g. [1 0] ) that are not in the range of T . T is said to be injective (or one-to-one ) if for all distinct x, y ? V, T(x) ? T(y) . In casual terms, it means that different inputs lead to different outputs. If T is injective, it is called an injection .