Question: How do you know if a transformation is injective or surjective?

How do you know if a transformation is injective?

A linear transformation is injective if the only way two input vectors can produce the same output is in the trivial way, when both input vectors are equal.

What is Injective and Surjective in linear transformation?

Theorem. If V and W are finite-dimensional vector spaces with the same dimension, then a linear map T : V → W is injective if and only if it is surjective. In particular, ker(T) = {0} if and only if T is bijective. … To find an isomorphism from a vector space V of dimension n to F, choose some basis v1,…,vn of V .

How do you tell if a matrix is surjective or injective?

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:

  1. injective if m≥n=rankA, in that case dimkerA=0;
  2. surjective if n≥m=rankA;
  3. bijective if m=n=rankA.

Can a transformation be injective but not surjective?

(Fundamental Theorem of Linear Algebra) If V is finite dimensional, then both kerT and R(T) are finite dimensional and dimV = dim kerT + dimR(T). If dimV = dimW, then T is injective if and only if T is surjective.

Is T an injective linear map?

2. Let T:V→W T : V → W be a linear map between vector spaces. Then: T is injective⟺Ker(T)={0V}.

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Is an isomorphism injective?

A linear transformation T from a vector space V to a vector space W is called an isomorphism of vector spaces if T is both injective and surjective.