1. No category

Linear Transformation (p.63)
• Matrix-vector multiplication is a linear transformation.
• If 𝐴 is a 𝑚 × 𝑛 matrix, and 𝑢 is 𝑛 × 1 vector, then matrix 𝐴 maps the
vector 𝑢 from ℝ𝑛 to the 𝑚 × 1 vector 𝑢′ in ℝ𝑚 .
ℝ𝑛
ℝ𝑚
𝑢
domain
𝐴
𝑢′
Co-domain
1
Linear Transformation (p.63)
• The space in which the vector 𝑢 exist is called “Domain” (ℝ𝑛 ).
• The space in which the vector 𝑢 is mapped to is called “Co-domain”
(ℝ𝑚 ).
• The subset of co-domain in which all ℝ𝑛 vectors are mapped into is
called “Range”.
𝑛
𝑚
𝐴
ℝ
ℝ
Range
𝑢
domain
𝑢′
Co-domain
2
Linear Transformation (p.63)
• Example:
𝑥
1 0
1 0 𝑥
• 𝐴 = 0 1 , 𝐴𝑢 = 0 1 𝑦 = 𝑦
0
0 0
0 0
• The matrix 𝐴 is a ℝ2 → ℝ3 mapping.
𝑦′
𝑧′
ℝ3
𝑦
𝐴: ℝ2 → ℝ3
ℝ2
𝑥
𝑥
𝑅𝑎𝑛𝑔𝑒 = 𝑦
0
𝑥′
3
Linearity condition (p.65)
• A transformation 𝑇 is linear if it meets these requirements for all 𝑢
and 𝑣:
𝑇 𝒖 + 𝒗 = 𝑇 𝒖 + 𝑇(𝒗)
𝑇 𝑐𝒖 = 𝑐𝑇(𝒖)
• From above implied that
𝑇 𝟎 =𝟎
𝑇 𝑐𝒖 + 𝑑𝒗 = 𝑐𝑇 𝒖 + 𝑑𝑇(𝒗)
4
Linear transformation theorem (p.71)
5
Linear Transformation (p.72)
6
2
Geometric linear transformation in ℝ (p.73-75)
7
2
Geometric linear transformation in ℝ (p.73-75)
8
2
Geometric linear transformation in ℝ (p.73-75)
9
2
Geometric linear transformation in ℝ (p.73-75)
10