Can a linear transformation not be injective?

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}.

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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.