Plan

• 2.2
• 2.3
• Learn whatever we don’t cover.

2.2 Matrix Representation of a linear transformation.

• Crash course on matrix representations from Fⁿ to Fⁿ.
• First we need to represent all vectors by vectors.
• Define ordered basis for any vector space. Both finite and infinite dimensional.
• We can write vectors with respect to this ordered basis and come up with the coordinate vector.
• Consider 4x³ + 2x² + x + 1 = 43 + 57( − 2 + x) + 26( − 2 + x)² + 4( − 2 + x)³ in P₃(R) with respect to the 2 basis.
• If {v_j} is a basis for B, then T(v_j) = ∑_i = 1^ma_ijw_i. The matrix is then A = (aij).
• Matrix representation of linear transformations. Lower bracket is domain and upper is codomain.
• The space of linear transformations is a vector space.
• The bracket operation is a linear transformation.

2.3 Compositiion

• The composition of linear transformations is linear.
• Whenever compostion makes sense:
  • $T(U\sb{1} + U\sb{2}) = TU\sb{1} + TU\sb{2}$ and (U₁ + U₂)T = U₁ + U₂T.
  • T(U₁U₂) = (TU₁)U₂.
  • TI = IT = T
  • a(U₁U₂) = (aU₁)U₂ = U₁(a_U2).
• These also holds for matrices.
• proof of matrix multiplication. In particular, the bracket is multiplicative.