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Recipe

Compute T^{*} on a specific functional

Use when: you're given T's action in coordinates and a functional \psi, and need T^{*}(\psi)=\psi\circ T as an explicit formula

Goal: an explicit formula for T^{*}(\psi), as a functional of the original domain's coordinates

Taught in The dual (transpose) map T*

Quick recipe

  1. Write T's action explicitly in coordinates
  2. Substitute T's output directly into \psi's formula
  3. Simplify to get T^{*}(\psi) as a single formula
  4. For a composite map, choose direct substitution or the reversed order
  1. 1

    Compute T(x,y) directly from T's matrix, writing both output coordinates as explicit expressions in x and y.

    Example

    For T=\begin{pmatrix}3&-1\\1&2\end{pmatrix}: T(x,y)=(3x-y,\ x+2y)

    Why: T^{*}(\psi)=\psi\circ T is a composition, and composing with T is meaningless until T's own output is written explicitly.

  2. 2

    Take \psi's formula and replace its own input coordinates with T's two output expressions from Step 1, in matching order.

    Example

    For \psi(w_1,w_2)=w_1-2w_2: (\psi\circ T)(x,y) = (3x-y) - 2(x+2y)

    Why: this substitution is the literal meaning of \psi\circ T, computing \psi at the point T actually sends (x,y) to.

  3. 3

    Expand and collect like terms, combining every x term and every y term separately.

    Example

    (3x-y)-2(x+2y) = 3x-y-2x-4y = x-5y

    Why: the simplified expression is precisely T^{*}(\psi), a genuine functional on V built entirely from T and \psi.

  4. 4

    To find (T_1T_2)^{*}(\psi), either apply Steps 1 through 3 to the composite map T_1T_2 directly, or apply them twice, first to T_1 then to T_2, composing the results in reversed order.

    • computing (T_1T_2)^{*}(\psi) directly — apply Steps 1 through 3 once, treating T_1T_2 itself as the map.
    • checking the order-reversal formula — apply Steps 1 through 3 to get T_1^{*}(\psi) first, then apply them again with T_2 in place of T_1, using T_1^{*}(\psi) as the new functional.

    Why: both routes compute the same functional, since (T_1T_2)^{*}=T_2^{*}T_1^{*}, so either path is a valid check on the other.

Common mistakes

  • Substituting \psi's coordinates into T instead of substituting T's output into \psi.
  • Forgetting that (T_1T_2)^{*}=T_2^{*}T_1^{*} reverses the order, and composing T_1^{*}T_2^{*} instead.
  • Simplifying only part of the expression, leaving x and y terms uncombined.
  • Treating T^{*}(\psi) as a vector instead of recognizing it as a new functional on V.