**Contents**show

## How do you know if a linear 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.

## Can a linear transformation be injective and 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 a linear transformation surjective?

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. If a linear transformation is an isomorphism and is defined by multiplication by a matrix, explain why the matrix must be square.

## How do you show that a linear transformation is not injective?

To test injectivity, one simply needs to see if the dimension of the kernel is 0. **If it is nonzero, then the zero vector and at least one nonzero vector have outputs equal 0W**, implying that the linear transformation is not injective.

## 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}**.

## How do you know if a function is Injective?

To show that a function is injective, we assume that there are **elements a1 and a2 of A with f(a1) = f(a2)** and then show that a1 = a2. Graphically speaking, if a horizontal line cuts the curve representing the function at most once then the function is injective.

## Can a matrix be injective but not 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).

## What is an injective matrix?

Let A be a matrix and let Ared be the row reduced form of A. **If Ared has a leading 1 in every column, then A is injective**. If Ared has a column without a leading 1 in it, then A is not injective. Invertible maps. If a map is both injective and surjective, it is called invertible.