Linear transformations on a plane

from University of Tokyo 1S1 Mathematical Sciences Foundation: Linear Algebra Linear transformations on a plane

• Since I didn’t take notes on my laptop at that time, I will only write down the key points and my thoughts as a review.
• Linear combinations of plane vectors:
  • For vectors $\overrightarrow{v_1},...,\overrightarrow{v_k}$,
    • Adding them together with weights ${\lambda }_1\overrightarrow{v_1}+{\lambda }_2\overrightarrow{v_2}+...+{\lambda }_k\overrightarrow{v_k}$ is called a linear combination.
    • It’s like adding them with weighted coefficients.
• Linear transformations on a plane:
  • Define the operation of matrix multiplication with vectors.
    • 
    • This definition is designed to make matrix multiplication and linear transformations on a plane equivalent.
      • I see, it seems like it’s designed as a practical tool. (blu3mo)
        • Well, it can also be seen as a generalization of linear transformations on a plane and multiplication in ℝ^1. (blu3mo)
          • That’s more interesting. (blu3mo)
    • In other words, a 2x2 matrix corresponds to a linear transformation in a one-to-one manner.
      • So it means it’s a bijection.
      • The matrix corresponding to a linear transformation is called a representation matrix.
    • In this case, the vector at the origin remains the same even after the linear transformation.
      • It’s obvious from the equation.
    • Terminology:
      • Identity matrix $\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$:
        • Equivalent to the number 1 in real numbers, as it doesn’t change anything when multiplied.
      • Zero matrix $\left[\begin{array}{cc}0& 0\\ 0& 0\end{array}\right]$:
        • Equivalent to the number 0 in real numbers, as it becomes 0 when multiplied by any value.
    • What can be done:
      • Scaling without changing the direction: $\left[\begin{array}{cc}\lambda & 0\\ 0& \lambda \end{array}\right]$
      • Reflection with respect to the y-axis: $\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$
      • Rotation: $\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$
        • This holds true when imagining what happens when θ is 0 or 90. (blu3mo)
        • It’s amazing how it aligns with the addition theorem. (blu3mo)
      • Is this related to Affine Transformation? (blu3mo)
        • It’s generally the same, but there are some differences.
          • https://www.geisya.or.jp/mwm48961/electro/transform2.htm#::text=写像のうち特に元,を１次変換という．&text=このような定数項,1次変換ではない.
          • Affine transformations can have constant terms and can perform parallel shifts.
            • Linear transformations cannot do that.
    • With this, we can say that linear transformations have linearity.
    • Furthermore, matrix multiplication is also defined.
      • This is done in a way that the linear transformations A(BV) and B(AV) are the same.
        • We want linearity in composite transformations.
      • Define the rules accordingly.
    • Impressions:
      • The notation for mappings is really about typing.
        • It’s like writing $F:R^2\to R^2$, similar to func f(_: R^2) -> R^2.
    • So linear and affine are equivalent. (blu3mo)
      • Well, that makes sense.