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- T: ℝⁿ → ℝᵐ, x ↦ Ax. Domain ℝⁿ, codomain ℝᵐ, range = set of all Ax (= Col A).
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- T is linear ⇔ T(u + v) = T(u) + T(v) and T(cu) = cT(u). Consequently T(0) = 0.
Memory trickQuick test: if T(0) ≠ 0, T is NOT linear.
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MCQ
Which map is NOT linear?
T(0) ≠ 0.
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- Standard matrix: A = [T(e₁) T(e₂) … T(eₙ)], the images of the identity columns.
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Find the standard matrix of the rotation of ℝ² by 90° counter-clockwise.
T(e₁) = (0, 1), T(e₂) = (−1, 0), so A = [[0, −1], [1, 0]].
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- Geometric maps: reflections, contractions/expansions, shears, projections (see the tables).
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- Onto ℝᵐ ⇔ columns span ℝᵐ ⇔ pivot in every row. One-to-one ⇔ T(x) = 0 has only the trivial solution ⇔ columns independent ⇔ pivot in every column.
Related real-exam questions
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T: ℝ³ → ℝ² can never be:
3 columns in ℝ² are dependent.
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