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Kernel and Image


Linear Algebra·Kernel & imageThe kernel sits in the domain. The image sits in the codomain. Drag v on the left, watch Av move on the right.
Domain · ℝ²input
ker A = {0}v
v=(1.5, 1)

03aScenarios

Full rankrank 2 · ker = {0}
Image is originrank 0

01MultiplicationA · v = Av

Av=
1.5
1
=
1·1.5 + 0·1
0·1.5 + 1·1
=
1.5
1
Top row of A dotted with v → first entry of Av. Same for bottom row.

Identity

rank 2 · det 1

The identity matrix maps every vector to itself. Drag v anywhere — Av follows exactly. The kernel is just the origin; the image is the entire codomain.
Learn more about full-rank maps · all three categories

02Properties

Rank A2dim ker A0dim im A2Determinant1v ∈ ker A?no (trivial)
dim ker + dim im = 0 + 2 = 2
▸Sweep0°
Codomain · ℝ²output
Av
Av=(1.5, 1)

03bScenarios

Image is a linerank 1

Key Terms
Getting Started — Dual Canvas Layout
Dragging v and Spotting the Kernel
Av and the Image Line on the Right
Preset Scenarios — Three Rank Categories
Full Rank: Nothing Collapses
Rank 1: a Kernel Line and an Image Line
Rank 0: Everything Goes to the Origin
Editing the Matrix A Directly
Sweep Playback — Tracing the Image of a Circle
Display Layer Toggles
Defining the Kernel and Image
Three Rank Cases for 2x2 Matrices
The Rank-Nullity Theorem
Related Concepts






Key Terms

Kernel (Null Space) — The set of all vectors vv in the domain such that Av=0Av = 0. The directions the matrix collapses to the origin.

Image (Column Space) — The set of all outputs AvAv as vv ranges over the domain. The reachable region in the codomain, equal to the span of the columns of AA.

Rank — The dimension of the image. For a 2×22 \times 2 matrix, rank is 0, 1, or 2.

Nullity — The dimension of the kernel. For a 2×22 \times 2 matrix, nullity is 0, 1, or 2.

Rank-Nullity Theorem — For any 2×22 \times 2 matrix AA, dim⁡(ker⁡A)+dim⁡(im A)=2\dim(\ker A) + \dim(\text{im } A) = 2. What gets collapsed plus what gets reached always equals the input dimension.

Nilpotent — A matrix with A2=0A^2 = 0. In rank-1 cases the kernel and image are the same line.

Getting Started — Dual Canvas Layout

The visualizer has two side-by-side canvases. The left is the domain (input space); the right is the codomain (output space). The matrix AA lives between them in the center column.

• Left canvas — drag the orange handle to place vector vv. A red dashed line marks the kernel of AA.
• Right canvas — shows AvAv in cyan. A green line marks the image of AA.
• Center — the four-cell matrix AA, an explanation of the current preset, a properties card with rank and determinant, and a sweep playback panel.

The fundamental relationship: every vector you place on the left has a counterpart on the right computed by AvAv. The kernel is the set of inputs that get sent to the origin; the image is the set of all reachable outputs.

Dragging v and Spotting the Kernel

Click and drag anywhere on the left canvas to set the position of vv. The right canvas updates instantly.

• Drag vv onto the red dashed line (when rank is 1) — AvAv collapses to the origin and a red dashed ring appears in the codomain. The properties card flips "v in ker A?" to "yes — collapses".
• Drag vv along the kernel line — AvAv stays at zero the whole way. The kernel is a whole line of vectors, not a single point.
• Drag vv off the kernel — AvAv jumps back to the image line, scaled by how far vv sits from the kernel.

When AA has rank 2, the kernel shows as a small ring around the origin labeled ker⁡A={0}\ker A = \{0\} — only the zero vector is annihilated. When AA is the zero map, the kernel fills the entire canvas in concentric rings: every direction is annihilated.

Av and the Image Line on the Right

The right canvas shows the codomain. Watch where AvAv lands as you drag vv on the left.

• Rank 2 — AvAv can be anywhere in the plane. No image line drawn; the entire plane is the image.
• Rank 1 — AvAv is confined to a single line through the origin. Drag vv in any direction; AvAv slides along that one green line.
• Rank 0 — AvAv is permanently at the origin no matter where vv is. A small green disk marks im A={0}\text{im } A = \{0\}.

The green image line spans the same direction as the columns of AA. That is the geometric meaning of "column space": stack the columns side by side, and they span the image.

Preset Scenarios — Three Rank Categories

The scenarios split across the left and right panels by rank category.

• Full rank (rank 2) — identity, rotate (with a dropdown for 30deg⁡30\deg through 270deg⁡270\deg), horizontal shear, and a generic rotate-and-stretch. Kernel is just the origin; image is the entire plane.
• Image is a line (rank 1) — project to x-axis, project to y=xy = x, project to the 30deg⁡30\deg line, outer product (oblique projection), and a nilpotent matrix where kernel and image coincide on the x-axis.
• Image is origin (rank 0) — the zero map. Everything collapses.

Each scenario triggers a brief explanation in the center column describing what makes it structurally interesting. The nilpotent case is especially instructive: ker⁡A=im A\ker A = \text{im } A, so applying AA twice annihilates everything.

Full Rank: Nothing Collapses

Four presets are invertible. The frozen pair below shows the rotation: on the left, the domain with its kernel; on the right, the codomain with its image.

Neither highlight has anything to draw. The kernel is just the origin — no non-zero vector is sent to zero — and the image is the entire codomain plane rather than a line. Every vv you drag produces an AvAv of the same length, turned but never flattened.
domain — kernel in redcodomain — image in greenker A = {0}vAv
Full rank (rotation), frozen

Neither canvas has a highlight to draw: the kernel is only the origin and the image is the whole plane. v and Av have the same length - turned, never flattened.

This is the case where the two canvases carry the least information, and that absence is itself the lesson: kernel and image only become *visible objects* when the map loses something.

In rank-nullity terms, operatornamerank+operatornamenullity=2operatorname{rank} + operatorname{nullity} = 2 is satisfied by 2+02 + 0. The image uses both available dimensions, so the kernel gets none. That accounting is what makes the map invertible — no two vectors share an image, so A−1A^{-1} exists and can undo it.

Rank 1: a Kernel Line and an Image Line

Five presets are singular with rank 1, and this is where both canvases finally have something to show. The frozen pair is the projection onto the x-axis.

On the left, a red kernel line: every vector along it is sent to the origin. On the right, a green image line: everything in the domain lands somewhere on it. Both are one-dimensional.
domain — kernel in redcodomain — image in greenker Avim AAv
Rank 1 (projection to x), frozen

Both lines finally appear - the red kernel line on the left, the green image line on the right. Everything in the domain lands somewhere on that one green line.

Drag vv onto the kernel line and AvAv collapses to a point at the origin on the right — the clearest demonstration in the tool that information is being destroyed. Drag vv anywhere else and AvAv moves along the image line but never leaves it.

Here rank-nullity reads 1+1=21 + 1 = 2, and the two canvases show the two halves of that sum simultaneously. Note the kernel and image need not be perpendicular, or even different lines: for the projection onto y=xy = x they are perpendicular, while the nilpotent preset has a kernel and image that coincide — the same line playing both roles, which is exactly what makes A2=0A^2 = 0.

Rank 0: Everything Goes to the Origin

One preset takes it to the limit. The zero matrix sends every vector to the origin, so the codomain canvas shows a single point and no image line at all.

On the left the kernel is not a line but the whole plane — shading everything, since there is no vector the map does not annihilate.
domain — kernel in redcodomain — image in greenker A = ℝ²vim A = {0}Av = 0
Rank 0 (the zero map), frozen

The codomain is a single point at the origin, and the kernel on the left is the entire plane rather than a line.

Rank-nullity still balances, at the extreme: 0+2=20 + 2 = 2. The image has dimension 0 and the kernel dimension 2.

Comparing this against the rank-1 case is the point of having it. Both are singular and both have det=0det = 0, but the projection keeps one dimension of information while the zero map keeps none. Rank distinguishes them; the determinant cannot. That is the practical argument for rank as the finer measure of what a matrix does.

Editing the Matrix A Directly

Card 01 (Multiplication) shows the equation Av=A⋅vAv = A \cdot v with the four entries of AA as editable input fields. Type a new value or use the arrow keys to step by 0.1.

• Change a top-row entry — the first component of AvAv updates. The expansion in the middle column shows the dot product explicitly: a11⋅v1+a12⋅v2a_{11} \cdot v_1 + a_{12} \cdot v_2.
• Set both columns parallel — rank drops to 1, the kernel line appears, the image collapses to a line. Watch the properties card switch from rank 2 to rank 1.
• Set all four entries to zero — rank is 0, the entire canvas becomes the kernel.

Editing the matrix breaks the connection to whichever preset was active, but you can always click a scenario to snap back to a canonical example.

Sweep Playback — Tracing the Image of a Circle

The sweep panel animates vv around a circle from 0deg⁡0\deg to 360deg⁡360\deg, leaving a fading trail in both canvases.

• Play — orange trail dots paint a circle on the left; cyan trail dots paint the image of that circle on the right.
• Step buttons — advance or rewind by 30deg⁡30\deg for frame-by-frame inspection.
• Scrub slider — drag to any angle; the trail clears when scrubbing.
• Reset — clear the trail and return vv to its starting angle.

The right-side trail reveals what AA does to circles: a rank-2 matrix maps a circle to an ellipse, a rank-1 matrix collapses the entire circle to a line segment, and a rank-0 matrix collapses it to a single point at the origin.

Display Layer Toggles

The chip strip above the matrix toggles which visual elements appear in both canvases.

• grid — standard integer grid in both spaces. Off for a cleaner background.
• kernel — the red dashed line (or ring) marking ker⁡A\ker A. Off to hunt for the kernel by dragging vv and watching for AvAv to vanish.
• image — the green line marking im A\text{im } A. Off to discover the image from sweep trails.
• trail — the fading dots left by sweep playback. Off for static snapshots.
• swarm — 140 sample points scattered across the domain, with their images on the codomain. Useful for seeing how the whole plane maps at once.
• labels — the vv, AvAv, ker⁡A\ker A, im A\text{im } A tags.

Combination tip: turn on swarm and turn off grid — the deformation of the plane by AA becomes vivid.

Defining the Kernel and Image

For a linear map A:Rn→RmA: \mathbb{R}^n \to \mathbb{R}^m, two natural subspaces emerge.

The kernel (or null space) is everything AA sends to zero:

ker⁡A={v∈Rn:Av=0}\ker A = \{v \in \mathbb{R}^n : Av = 0\}


This is a subspace of the domain. It captures what information AA destroys.

The image (or column space) is everything AA can produce:

im A={Av:v∈Rn}=span(columns of A)\text{im } A = \{Av : v \in \mathbb{R}^n\} = \text{span}(\text{columns of } A)


This is a subspace of the codomain. It captures what outputs are reachable.

For deeper treatment see kernel theory page, image of a linear map, and column space.

Three Rank Cases for 2x2 Matrices

A 2×22 \times 2 matrix AA falls into exactly one of three structural cases, classified by rank.

• Rank 2 (invertible) — det⁡(A)≠0\det(A) \neq 0. The kernel is trivial ({0}\{0\} only) and the image is all of R2\mathbb{R}^2. The map is one-to-one and onto.
• Rank 1 — det⁡(A)=0\det(A) = 0 but AA is not zero. The kernel is a line through the origin (one direction collapses) and the image is also a line through the origin (one direction reaches). The two lines can be perpendicular (orthogonal projection), at an angle (oblique projection), or identical (nilpotent).
• Rank 0 — A=0A = 0. The kernel is all of R2\mathbb{R}^2 and the image is just the origin. Every input gets annihilated.

The rank can be detected from the determinant alone for 2×22 \times 2 matrices: nonzero means rank 2, zero with at least one nonzero entry means rank 1, all zeros means rank 0.

For more see matrix rank, invertible matrices, and nilpotent matrices.

The Rank-Nullity Theorem

For any 2×22 \times 2 matrix AA, the dimensions of the kernel and image always add to 2:

dim⁡(ker⁡A)+dim⁡(im A)=2\dim(\ker A) + \dim(\text{im } A) = 2


More generally, for a matrix with nn columns,

nullity(A)+rank(A)=n\text{nullity}(A) + \text{rank}(A) = n


The intuition: every input direction either gets annihilated (counts toward the kernel) or makes it through (counts toward the image). There is no third category.

Reading off the visualizer: the properties card prints the rank-nullity equation live for the current AA. A rank-2 matrix has 0+2=20 + 2 = 2; a rank-1 matrix has 1+1=21 + 1 = 2; the zero matrix has 2+0=22 + 0 = 2. The total never changes.

For comprehensive coverage see rank-nullity theorem, dimension formula, and fundamental theorem of linear maps.

Related Concepts

Column Space — same as the image; spanned by the columns of AA.

Null Space — same as the kernel; solutions to Av=0Av = 0.

Rank of a Matrix — the dimension of the image, equal to the number of linearly independent columns.

Determinant — for a 2×22 \times 2 matrix, det⁡(A)=0\det(A) = 0 exactly when the matrix has nontrivial kernel.

Linear Independence — the columns of AA are linearly independent if and only if AA has rank 2.

Invertible Matrices — the case where kernel is trivial and image is the full codomain.

Row Space and Four Fundamental Subspaces — kernel and image of AA together with kernel and image of ATA^T form the four fundamental subspaces of linear algebra.

Rank-Nullity Theorem — the dimension equation that ties kernel and image together.