Linear Transformations

If a transformation on a vector is linear then it can be done with a matrix. If a transformation on a vector can be done with a matrix, then it is linear.

Let v and w be vectors that can be added together to make a third vector that we call “v+w”.

Let M be a matrix that can operate on these three vectors. The operation is matrix multiplication.

  • Let M(v) be the vector we get if M operates on v.
  • Let M(w) be the vector we get if M operates on w.
  • Let M(v+w) be the vector we get if M operates on the “v+w” vector.

M(v+w) = M(v) + M(w)

Let a be a scalar.

Let av be the vector we get from multiplying v by a.

M(av) = a M(v)

When you look at the two equations below, you see the definition of Linear:

  • M(v+w) = M(v) + M(w)
  • M(av) = a M(v)

As an alternative

  • M(av + bw) = aM(v) + bM(w)

The above equation proves that m is a linear transformation because we see both additivity and scalar multiplication in it.

Appendix A – Example

\begin{bmatrix}a_{11} && a_{12} \\ a_{21} && a_{22} \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \begin{bmatrix} a_{11} v_1 + a_{12} v_2 \\ a_{21} v_1 + a_{22} v_2 \end{bmatrix}

\begin{bmatrix}a_{11} && a_{12} \\ a_{21} && a_{22} \end{bmatrix} \begin{bmatrix} w_1 \\ w_2 \end{bmatrix} = \begin{bmatrix} a_{11} w_1 + a_{12} w_2 \\ a_{21} w_1 + a_{22} w_2 \end{bmatrix}

\begin{bmatrix}a_{11} && a_{12} \\ a_{21} && a_{22} \end{bmatrix} \begin{bmatrix} v_1 + w_1 \\ v_2 + w_2 \end{bmatrix} = \begin{bmatrix} a_{11} (v_1 + w_1) + a_{12} (v_2 + w_2) \\ a_{21} (v_1 + w_1) + a_{22} (v_2 + w_2) \end{bmatrix}

\begin{bmatrix} a_{11} (v_1 + w_1) + a_{12} (v_2 + w_2) \\ a_{21} (v_1 + w_1) + a_{22} (v_2 + w_2) \end{bmatrix} = \begin{bmatrix} a_{11} v_1 + a_{11} w_1 + a_{12} v_2 + a_{12} w_2 \\ a_{21} v_1 + a_{21} w_1 + a_{22} v_2 + a_{22} w_2 \end{bmatrix}

The last matrix shows that A operating on v+w has all the terms of A operating on v and A operating on w — A(v+w) = A(v) + A(w).

\begin{bmatrix}a_{11} && a_{12} \\ a_{21} && a_{22} \end{bmatrix} \begin{bmatrix} kv_1 \\ kv_2 \end{bmatrix} = \begin{bmatrix} a_{11} k v_1 + a_{12} k v_2 \\ a_{21} k v_1 + a_{22} k v_2 \end{bmatrix} = \begin{bmatrix} k a_{11} v_1 + k a_{12} v_2 \\ k a_{21} v_1 + k a_{22} v_2 \end{bmatrix}

k \begin{bmatrix}a_{11} && a_{12} \\ a_{21} && a_{22} \end{bmatrix} \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = k \begin{bmatrix} a_{11} v_1 + a_{12} v_2 \\ a_{21} v_1 + a_{22} v_2 \end{bmatrix} = \begin{bmatrix} k a_{11} v_1 + k a_{12} v_2 \\ k a_{21} v_1 + k a_{22} v_2 \end{bmatrix}