# 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 {vj} is a basis for B, then T(vj) = ∑i = 1maijwi. 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(aU2).
• These also holds for matrices.
• proof of matrix multiplication. In particular, the bracket is multiplicative.