Math · Vectors

Matrix Transformations / Stretch, Shear, Reflection

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Wrap-up

What you learned

Recommended next: Open concept testCheck whether the core ideas are ready without leaving this concept.
Read next: Dot Product / Angle and ProjectionKeep the vectors branch moving into alignment and projection

Key takeaway

The two matrix columns are the images of the two basis vectors.
Shear, stretch, and reflection are visible whole-plane actions, not separate tricks.
A sample point or triangle is evidence for the same linear rule that moves the grid.

Common misconception

Do not start with one isolated point and ignore the basis arrows; the columns are the cleaner first reading.

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

Keep this idea moving

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Keep the vectors branch moving into alignment and projectionVectors

Dot Product / Angle and Projection

Keep two vectors, their angle, the signed projection of one onto the other, and the dot product visible together so alignment reads geometrically instead of as memorized cases.

Reuse the same plane for scale-and-turn actions on complex numbersComplex Numbers and Parametric Motion

Complex Numbers on the Plane

Read complex numbers as points and vectors on one plane, then keep addition and multiplication geometric instead of symbolic-only.

Next in Combination and scalingVectors

Vectors in 2D

Combine, subtract, and scale vectors on one plane so magnitude, direction, and components stay tied to the same live object.

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Reference and support

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Equation snapshotColumn-first map · Columns are basis imagesShow 3 equations

Read the columns before tracking an individual point.

Column-first map
$M = [a c b d]$
Records one linear rule that acts on every point in the plane.
Columns are basis images
$M e_{1} = ⟨ a, c ⟩, M e_{2} = ⟨ b, d ⟩$
The columns are the images of the two basis vectors, so they organize the transformed grid.
Action on any vector
$M ⟨ x, y ⟩ = ⟨ a x + b y, c x + d y ⟩$
Any vector is rebuilt from the same basis-image columns.

Why it behaves this way

Explanation

A 2 by 2 matrix is easiest to read as a rule that transforms the whole plane, not as a detached box of numbers. This bench keeps the grid, basis vectors, unit square, and sample triangle on one coordinate system so stretch, shear, reflection, and mixed actions stay visible together.

Read the columns first. The first column is $M e_{1}$ and the second column is $M e_{2}$ . Once you know where those two basis vectors land, every other point follows by the same linear-combination rule, which is why the entire grid moves consistently.

Key ideas

01The two matrix columns tell you where

e_{1}

and

e_{2}

land, so they already predict the new grid and the image of the unit square.

02Every vector is built from x copies of

e_{1}

and y copies of

e_{2}

, so after transformation it becomes the same combination of

M e_{1}

and

M e_{2}

03Stretch, shear, and reflection are different plane-wide outcomes of the same linear rule.

Worked examples

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Frozen walkthrough

Step through the frozen example

Frozen walkthrough

Use one shear case and one reflection case. In each example, read the columns first, then check the same story on the plane.

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Example 1 of 2

Frozen valuesUsing frozen parameters

For the shear matrix $M = [1101]$ , where do $e_{1}$ and $e_{2}$ land, and what does that predict for the unit square?

a

First-column x entry

c

First-column y entry

b

Second-column x entry

d

Second-column y entry

1. Read the two columns

The matrix columns are

⟨ 1, 0 ⟩

and

⟨ 1, 1 ⟩

2. Match the first column to $M e_1$

Because

e_{1} = ⟨ 1, 0 ⟩

, the first column is its image, so

M e_{1} = ⟨ 1, 0 ⟩

3. Match the second column to $M e_2$

Because

e_{2} = ⟨ 0, 1 ⟩

, the second column is its image, so

M e_{2} = ⟨ 1, 1 ⟩

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.

Quick test

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Accessibility

Open the text-first descriptions when you need the simulation and graph translated into words.

The simulation shows a coordinate plane with a transformed grid, two draggable basis-image vectors, a transformed unit square, a transformed sample triangle, and a tracked point. Sliders control the four matrix entries that determine where the two basis vectors land.

Changing an entry updates the basis images, the grid, the unit square, and the sample triangle together so the learner can connect the matrix columns to one plane-wide action.

Graph summary

One graph tracks the x- and y-coordinates of a fixed 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 picture, so hovering them previews an intermediate transformation directly on the grid and shapes.

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Open a reflection and compare the original and transformed triangle.

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