Matrix-Vector Transformations

Lesson 11 of 12: VM Cluster

In this lesson:

• Write vectors as column matrices and compute
• Interpret matrix-vector products as geometric transformations
• Compose transformations using matrix multiplication

Matrices Transform Every Image on Screen

Every pixel on a screen has coordinates — a position vector.

Zooming in: each position vector is scaled.

Rotating: each position vector is rotated.

Reflecting: each position vector is flipped.

One matrix encodes one transformation — applied to every vector at once.

Rewriting Vectors as Column Matrices

A vector can be written as a column matrix:

The Row-Dot Rule for Matrix-Vector Products

() times () gives a result.

Each row of dotted with gives one entry.

Computing : Step by Step

,

Row 1:

Row 2:

Off-Diagonal Entries Mix the Components

Wrong: — diagonal only

Correct: — full dot products

Off-diagonal entries and mix and — that's what makes transformations interesting.

Moves a Vector: It's a Transformation

You computed where landed: from to .

The matrix is a transformation: it maps every input vector to an output vector.

Every point in the plane gets moved — the matrix describes the movement.

Scaling: One Matrix Stretches Every Vector

• (doubled)
• (doubled)
• (doubled)

Every vector doubles in length; direction unchanged.

Reflection: Flipping Across an Axis

• ( negated)
• ( negated)
• -coordinates are unchanged; -coordinates flip sign

Rotation 90°: Turning the Plane

• (right → up)
• (up → left)

Every vector rotates 90° counterclockwise.

What Every Linear Transformation Preserves

Under any transformation :

• Origin stays at origin: always
• Lines through origin remain lines:
• Parallel lines stay parallel

Not preserved in general: lengths, angles, areas.

Identify Each Transformation from Its Matrix

Apply to unit vectors to identify each transformation:

Quick Check: Apply and Identify

1. , : compute

2. , : compute

3. Name the transformation:

One Matrix Encodes Two Transformations

Two steps: apply , then apply .

— this is a composition.

Can we find one matrix such that ?

Yes: — the product of the two matrices.

Composition: Rightmost Factor Applies First

To apply first, then : compute .

Mnemonic: "rightmost matrix applies first" — " then " =

Computing a Composite Transformation Step by Step

(scale 2) then (reflect -axis): composite =

Verify: , — matches

Order Matters for Non-Commuting Transforms

(rotation), (shear)

Order changes the result.

Write the Composite Matrix Yourself

"First rotate 90° CCW, then reflect over the -axis."

1. Write (rotation) and (reflection)
2. Identify which is first — write it on the right
3. Compute the composite matrix
4. Apply to

No scaffolding — all steps on your own.

Spot and Fix the Order Error

A student wrote: "Apply then " . Find the error.

, ,

Compute and . Which matches "A first, then B"?

What a 2×2 Matrix Really Encodes

A matrix is a transformation of the plane:

• The columns of show where and land
• Matrix product = composition of transformations
• Order matters — non-commutativity has geometric meaning

Every matrix computation you've done since VM.C.8 was a transformation.

Area Scaling and the Determinant Preview

Every matrix scales areas by .

Next lesson (VM.C.12): explore what the determinant means geometrically — and which transformations are invertible (reversible).

• : areas expand
• : areas preserved (rotation, reflection)
• : plane collapses — not invertible