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

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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.

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Stable links

Why it behaves this way

Explanation

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.

Frozen walkthrough

Step through the frozen example

Frozen walkthrough

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

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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.

Make a prediction before you reveal the next step.

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

Check your reasoning against the live bench.

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?

Use the live bench to test the result before moving on.

Accessibility

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 this idea moving

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Keep the vectors branch moving into alignment and projectionVectors

Matrix Transformations / Stretch, Shear, Reflection

Explanation

Step through the frozen example

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 this idea moving

Dot Product / Angle and Projection

Complex Numbers on the Plane

Vectors in 2D

Matrix Transformations / Stretch, Shear, Reflection

Explanation

Step through the frozen example

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 this idea moving

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?