Linear Algebra Lesson

Matrix of a Linear Transformation: From Rules to Columns

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11th-12th Grade (Advanced)BTC Style / Thinking ClassroomsCCSS.MATH.CONTENT.HSN-VM.C.6,CCSS.MATH.CONTENT.HSN-VM.C.850 min

Learning Objective

Students will construct the standard matrix of a linear transformation by applying T to each standard basis vector and assembling the resulting images as columns, then use that matrix to evaluate T on arbitrary vectors.

Lesson Overview

Students discover that any linear transformation T: ℝⁿ → ℝᵐ is completely determined by where it sends the standard basis vectors, and that those images form the columns of the transformation matrix. Tasks progress from evaluating T at specific vectors to constructing the full matrix and applying it to new inputs.

Materials

Vertical whiteboards or chart paper
Dry-erase markers (multiple colors)
Printed reference card: standard basis vectors e₁, e₂, e₃
Random grouping tool

Scaffolded Task Progression

1Task 1

A linear transformation

T : R^{2} \to R^{2}

is defined by the rule

T ([x y]) = [3 x 2 y] .

Compute each output:(a)

T ([10])

,(b)

T ([01])

,(c)

T ([4 - 1])

2Task 2

Look at your answers from Task 1. Arrange

T (e_{1})

and

T (e_{2})

side by side as columns of a

2 \times 2

matrix

A

. Now compute

A [4 - 1]

using matrix-vector multiplication. What do you notice?

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