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 ,
- Adding them together with weights is called a linear combination.
- It’s like adding them with weighted coefficients.
- For vectors ,
- 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)
- Well, it can also be seen as a generalization of linear transformations on a plane and multiplication in ℝ^1. (blu3mo)
- I see, it seems like it’s designed as a practical tool. (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 :
- Equivalent to the number 1 in real numbers, as it doesn’t change anything when multiplied.
- Zero matrix :
- Equivalent to the number 0 in real numbers, as it becomes 0 when multiplied by any value.
- Identity matrix :
- What can be done:
- Scaling without changing the direction:
- Reflection with respect to the y-axis:
- Rotation:
- 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.
- https://www.geisya.or.jp/
- It’s generally the same, but there are some differences.
- 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.
- This is done in a way that the linear transformations A(BV) and B(AV) are the same.
- Impressions:
- The notation for mappings is really about typing.
- It’s like writing , similar to
func f(_: R^2) -> R^2
.
- It’s like writing , similar to
- The notation for mappings is really about typing.
- So linear and affine are equivalent. (blu3mo)
- Well, that makes sense.
- This definition is designed to make matrix multiplication and linear transformations on a plane equivalent.
- Define the operation of matrix multiplication with vectors.