Image (linear map)

The image of a linear map $T:V\to W$ is the set of all vectors that occur as $T(v)$ for some $v\in V$. It is denoted by $\operatorname{im}T$, $\operatorname{Im}T$, or $T(V)$, and is also called the range. Unlike the codomain $W$, which specifies the space into which the map takes values, the image contains only the values actually attained. In linear algebra, the image describes the attainable outputs of a transformation and connects its algebraic structure with matrix rank and solvability. (math.libretexts.org)

Definition and subspace structure

Let $V$ and $W$ be vector spaces over the same field $F$. The image of $T$ is

It is always a linear subspace of $W$. Indeed, $T(0)=0$, so the image contains the zero vector. If $y_1=T(v_1)$, $y_2=T(v_2)$, and $a,b\in F$, then

Thus every linear combination of image vectors belongs to the image. This property distinguishes images of linear maps from images of arbitrary functions, which need not have a vector-space structure. (math.libretexts.org)

A map is a surjective function precisely when $\operatorname{im}T=W$. Every linear map becomes surjective if its codomain is restricted to its image. Conversely, the kernel consists of input vectors sent to zero:

The image lies in the output space, whereas the kernel lies in the input space. (math.libretexts.org)

Spanning sets and matrix representation

If $v_1,\ldots,v_n$ is a basis of a finite-dimensional space $V$, then

To see this, write an arbitrary input as a linear combination of the basis vectors and apply linearity. The images of the basis vectors therefore generate the entire image, although they may be zero or fail to be linearly independent. An independent subset spanning the same space provides a basis of the image. The same principle applies to any spanning set of $V$. (ximera.osu.edu)

After bases are chosen, $T$ is represented by an $m\times n$ matrix $A$. For the associated map $x\mapsto Ax$, its image is the column space:

where $a_j$ are the columns of $A$. Thus its outputs form the linear span of the columns. For an abstract map, these columns represent the coordinates of $T(v_j)$; the column space is consequently the coordinate representation of the image. (math.purdue.edu)

Rank and computation

The dimension of the image is the rank of $T$. In matrix coordinates it equals the rank of the representing matrix. If $V$ is finite-dimensional, the rank–nullity theorem states

It measures how the input dimension divides between directions annihilated by the map and independent output directions. In particular, the image cannot have greater dimension than the domain. (ximera.osu.edu)

A basis for a matrix image can be found using Gaussian elimination. Reduce $A$ to row-echelon form, identify its pivot-column positions, and select the corresponding columns of the original matrix. These columns form a basis of $\operatorname{im}A$. The distinction matters: row operations generally change the column space, even though they preserve the dependence relations needed to identify which original columns form a basis. (ocw.mit.edu)

Images and linear equations

A system of linear equations

has a solution exactly when $b\in\operatorname{im}A$. Image membership is therefore the condition for consistency. If $x_0$ is one solution, all solutions are

The image determines which right-hand sides are attainable; the kernel, also called the matrix null space, determines the freedom among inputs attaining a given right-hand side. (math.purdue.edu)

For example, consider

Every output has the form $t(1,2)$, and every such vector is attained by taking $(x,y,z)=(t,0,0)$. Hence

The image is a line rather than the whole codomain. Accordingly, $T(x,y,z)=(b_1,b_2)$ is solvable exactly when $b_2=2b_1$, illustrating the image criterion for consistency. (math.purdue.edu)

Quotient-space interpretation

The image has a canonical description using the quotient vector space $V/\ker T$. The rule

defines an isomorphism. It is well-defined because inputs differing by a kernel vector have the same output, and it is both injective and surjective. This is the first isomorphism theorem for vector spaces:

The quotient identifies precisely those inputs that the map cannot distinguish. Each resulting class corresponds to one attained output, making the image the vector space obtained after this input redundancy is removed. This description remains valid without a finite-dimensional assumption. (homepages.math.uic.edu)