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## How do you tell if a linear transformation is surjective?

A transformation T mapping V to W is called surjective (or onto) if **every vector w in W is the image of some vector v in V**. [Recall that w is the image of v if w = T(v).] Alternatively, T is onto if every vector in the target space is hit by at least one vector from the domain space.

## Is a linear transformation 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.

## Is a linear transformation bijective?

A linear transformation can be **bijective only if its domain and co-domain space have the same dimension**, so that its matrix is a square matrix, and that square matrix has full rank.

## 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**.

## How do you check for Surjectivity?

A function f (from set A to B) is surjective if and only if for every y in B, there is at **least one x in A such that f(x) = y**, in other words f is surjective if and only if f(A) = B.

## Is linear functional injective?

A linear transformation **is injective if and only if its kernel is the trivial subspace {0}**. Example. This is completely false for non-linear functions. For example, the map f : R → R with f(x) = x2 was seen above to not be injective, but its “kernel” is zero as f(x)=0 implies that x = 0.

## How do you know if a linear transformation is 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. Conversely, assume that ker(T) has dimension 0 and take any x,y∈V such that T(x)=T(y).

## Is a transformation linear?

A linear transformation is **a function from one vector space to another that respects the underlying (linear) structure of each vector space**. A linear transformation is also known as a linear operator or map. … The two vector spaces must have the same underlying field.

## How do you know if a linear transformation is onto?

Let T:Rn↦Rm be a linear transformation. Then T is called onto **if whenever →x2∈Rm there exists →x1∈Rn such that T(→x1)=→x2**. We often call a linear transformation which is one-to-one an injection. Similarly, a linear transformation which is onto is often called a surjection.

## Are linear maps bijective?

Definition A linear map T : V → W is called **bijective if T is both injective and surjective**. Let T : V → W be a linear map. Then T is injective if and only if null(T) = {0}.