Concept module

Matrix Transformations / Stretch, Shear, Reflection

Let one 2 by 2 matrix act on a grid, the basis vectors, and a sample shape so stretch, shear, reflection, and combined plane changes stay visual instead of symbolic-only.

Interactive lab

Keep the stage, immediate control feedback, and linked study tools in one working bench.

Matrix transformations

Drag the images of e1 and e2 or edit the matrix entries directly. The grid, the unit square, and the sample triangle all follow the same 2 by 2 action on the plane.

columns = images of e1 and e2

Controls

Adjust the live parameters and watch the bench respond.

First-column x entry

a

Move the image of $e_1$ left or right.

First-column y entry

c

Move the image of $e_1$ up or down.

Second-column x entry

b

Move the image of $e_2$ left or right.

Second-column y entry

d

Move the image of $e_2$ up or down.

More tools

Secondary controls, alternate presets, and less-used toggles stay nearby without crowding the main bench.

Show

More presets

Presets

Predict -> manipulate -> observe

Keep the active prompt next to the controls so each change has an immediate visible consequence.

ObservationPrompt 1 of 3

Drag one basis-image handle and watch the same column change in the matrix, the grid, the square, and the sample triangle all at once.

Graphs

Switch graph views without breaking the live stage and time link.

Tracked point during the blend

Blend from the identity matrix to the current matrix and watch the tracked point coordinates evolve.

blend t: 0 to 1coordinate value: -4 to 4

probe xprobe y

Hover or scrub to link the graph back to the stage.blend t / coordinate value

Equation map

See each variable before you move it.

Select a symbol to highlight the matching control and the graph or overlay it most directly changes.

a

First-column x entry

Sets the x-component of $M e_1$.

Graph: Tracked point during the blendGraph: Basis-vector lengths during the blendOverlay: Basis vectorsOverlay: Unit square

Equations in play

Choose an equation to sync the active symbol, control highlight, and related graph mapping.

More tools

Detailed noticing prompts, guided overlays, and challenge tasks stay available without taking over the main bench.

Hide

What to notice

Keep the matrix entries tied to the visible plane action instead of treating them as detached numbers.

ObservationPrompt 1 of 3

Drag one basis-image handle and watch the same column change in the matrix, the grid, the square, and the sample triangle all at once.

Control: First-column x entryControl: First-column y entryControl: Second-column x entryControl: Second-column y entryGraph: Basis-vector lengths during the blendOverlay: Basis vectorsOverlay: Unit squareEquation

a

Equation

b

Equation

c

Equation

d

Guided overlays

Focus one overlay at a time to see what it represents and what to notice in the live motion.

4 visible

Overlay focus

Basis vectors

Show the original basis directions and their images.

What to notice

The columns are the new basis directions that organize the transformed plane.

Why it matters

The basis images are the fastest route from matrix entries to geometry.

Control: First-column x entryControl: Second-column x entryControl: First-column y entryControl: Second-column y entryGraph: Basis-vector lengths during the blendEquation

a

Equation

b

Equation

c

Equation

d

Challenge mode

Use the basis columns, the unit square, and the transformed sample shape together. This checkpoint is about reading a matrix as one plane action instead of four disconnected entries.

0/1 solved

MatchCoreSolved

8 of 8 checks

Right-shear checkpoint

Start from the identity matrix and build a right shear that keeps the first basis vector near its original x-axis direction while the unit square becomes a right-leaning parallelogram. Keep the basis and square visible so the column story stays on the plane.

Graph-linkedGuided start2 hints

Suggested start

Change the second column first. A right shear comes from moving where

e_{2}

lands while leaving

e_{1}

almost unchanged.

Matched

Open the Tracked point during the blend graph.

Tracked point during the blend

Matched

Keep the Basis vectors visible.

Matched

Keep the Unit square visible.

Matched

Keep the Sample triangle visible.

Matched

Keep the first-column x entry near

1

Matched

Keep the first-column y entry near

0

Matched

Push the second-column x entry close to

1

e_{2}

leans right.

Matched

Keep the second-column y entry near

1

so the shear does not collapse the vertical scale.

Challenge solved

You built the shear from the basis columns instead of treating the entries as disconnected numbers. The grid, square, and sample shape now lean together under one linear action.

M e1 = <1, 0>, M e2 = <1, 1>, and the tracked probe lands at <2.7, 1.1>. The transformed square keeps its orientation, so the action stays on the non-reflecting side. Off-axis entries are leaning the basis vectors, so the grid is shearing as well.

Equation detailsDeeper interpretation, notes, and worked variable context.

2 by 2 matrix

Packages one linear action on the plane into four entries.

M = [a c b d]

a

First-column x entry 1

c

First-column y entry 0

b

Second-column x entry 1

d

Second-column y entry 1

Columns are basis images

Explains why the two columns already tell you how the unit square and the whole grid will move.

M e_{1} = ⟨ a, c ⟩, M e_{2} = ⟨ b, d ⟩

a

First-column x entry 1

c

First-column y entry 0

b

Second-column x entry 1

d

Second-column y entry 1

Action on any vector

Shows how every point is rebuilt from the same basis-image columns.

M ⟨ x, y ⟩ = ⟨ a x + b y, c x + d y ⟩

a

First-column x entry 1

c

First-column y entry 0

b

Second-column x entry 1

d

Second-column y entry 1

Progress

Not startedMastery: NewLocal-first

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Let the live model runChange one real controlOpen What to notice

Try this setup

Jump to a named bench state or copy the one you are looking at now. Shared links reopen the same controls, graph, overlays, and compare context.

This bench opened from a setup link. Any new copy will reflect the state you see now.

Current bench

Shear right preset

This bench still matches one named preset, so the copied link will reopen that same starting point along with the current graph, overlays, and inspect context.

Open default bench

Featured setups

Shear right

Open a clean shear case with the basis vectors and square visible.

Reflect across y-axis

Open the reflection case and compare the original and transformed triangle.

Saved setups

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Checking saved setup access.

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Copy current setup

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Stable links

Short explanation

What the system is doing

A 2 by 2 matrix becomes easier to trust when it is treated as a visible action on the plane instead of a detached table of numbers. This bench keeps the transformed grid, the basis vectors, the unit square, and one sample triangle on the same coordinate system so stretch, shear, reflection, and combined actions stay geometric.

The cleanest reading is column-first. The first column tells you where $e_{1}$ lands, the second column tells you where $e_{2}$ lands, and every other point follows by the same combination rule.

Key ideas

01The columns of the matrix are the images of the basis vectors, so they already tell you how the unit square and the grid will lean, stretch, or reflect.

02A matrix acts on every point with the same linear rule $M\langle x, y \rangle = \langle ax + by, cx + dy \rangle$, so straight grid lines stay straight.

03Stretch, shear, and reflection are different visible outcomes of the same 2 by 2 action.

Worked example

Read the full frozen walkthrough.

Frozen walkthrough

Use two fixed matrix cases to keep the column rule tied to a clear plane picture.

Live worked examples are available on Premium. You can still read the full frozen walkthrough on the free tier.

View plans

Frozen valuesUsing frozen parameters

For the shear matrix $M = [1011]$ , where do $e_{1}$ and $e_{2}$ land?

a

First-column x entry

c

First-column y entry

b

Second-column x entry

d

Second-column y entry

1. Read the matrix columns directly

The matrix columns are

⟨ 1, 0 ⟩

and

⟨ 1, 1 ⟩

2. Apply it to the first basis vector

Because

e_{1} = ⟨ 1, 0 ⟩

, the first column is its image, so

M e_{1} = ⟨ 1, 0 ⟩

3. Apply it to the second basis vector

Because

e_{2} = ⟨ 0, 1 ⟩

, the second column is its image, so

M e_{2} = ⟨ 1, 1 ⟩

Current basis images

M e_{1} = ⟨ 1, 0 ⟩, M e_{2} = ⟨ 1, 1 ⟩

The first basis direction stays put while the second basis direction leans right, so the unit square becomes a right-leaning parallelogram.

Common misconception

Each entry moves the sample shape independently, so the grid and basis vectors are just decoration.

The entries matter because they set the two basis-image columns.

Once those columns are fixed, the grid, the unit square, and every sample point all follow from the same linear combination rule.

Mini challenge

Set the matrix so the unit square leans to the right without flipping over, and keep the first basis vector close to the original x-axis direction.

Prediction prompt

Decide whether you need a stretch, a shear, or a reflection before you touch the columns.

Check your reasoning

You need a positive-orientation shear: keep

M e_{1}

near

⟨ 1, 0 ⟩

while moving

M e_{2}

to the right.

That creates a slanted parallelogram instead of a mirror image. The basis columns control the whole picture.

Quick test

Reasoning

Question 1 of 3

Answer from the live plane picture and the column rule together.

What is the most honest first interpretation of the two columns of a 2 by 2 matrix on this bench?

Choose one answer to reveal feedback, then test the idea in the live system if a guided example is available.

Accessible description

The simulation shows a coordinate plane with the transformed grid, two draggable basis-image vectors, the transformed unit square, a transformed sample triangle, and a tracked probe point. Sliders change the four matrix entries that set the images of the two basis vectors.

Compare mode can overlay a second transformed setup, and graph hover can preview the same matrix action blended in from the identity matrix to the current matrix.

Graph summary

One graph tracks the x- and y-coordinates of a fixed probe point while the current matrix is blended in from the identity matrix. The second graph tracks the lengths of the transformed basis vectors over the same blend.

Those graphs are tied to the same plane, so hovering them previews an intermediate matrix action directly on the grid and shapes.

Keep moving through the concept map.

These suggestions come from the concept registry, so the reason label reflects either curated guidance or the fallback progression logic.

Keep the vectors branch moving into alignment and projectionVectors

Matrix Transformations / Stretch, Shear, Reflection

See each variable before you move it.

Basis vectors

Right-shear checkpoint

What the system is doing

Read the full frozen walkthrough.

For the shear matrix $M = [1011]$ , where do $e_{1}$ and $e_{2}$ land?

What is the most honest first interpretation of the two columns of a 2 by 2 matrix on this bench?

Keep moving through the concept map.

Dot Product / Angle and Projection

Complex Numbers on the Plane

Vectors in 2D

Matrix Transformations / Stretch, Shear, Reflection

See each variable before you move it.

Basis vectors

Right-shear checkpoint

What the system is doing

Read the full frozen walkthrough.

For the shear matrix M=[10​11​], where do e1​ and e2​ land?

What is the most honest first interpretation of the two columns of a 2 by 2 matrix on this bench?

Keep moving through the concept map.

Dot Product / Angle and Projection

Complex Numbers on the Plane

Vectors in 2D

For the shear matrix $M = [1011]$ , where do $e_{1}$ and $e_{2}$ land?