7-6.md

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title: 7/6

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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 $F^n$ to $F^n$.

• 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 $4x^3+2x^2+x+1=43 + 57 (-2 + x) + 26 (-2 + x)^2 + 4 (-2 + x)^3$ in $P_3(R)$ with respect to the 2 basis.

• If $\{v_j\}$ is a basis for $B$, then ParseError: KaTeX parse error: Expected group after '_' at position 12: T(v_j)=\sum_̲_{i=1} ^m a_{ij…. 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:

  • ParseError: KaTeX parse error: Undefined control sequence: \sb at position 4: T(U\̲s̲b̲{1} + U\sb{2}) … and $(U_1+U_2)T=U_1+U_2T$.

  • $T(U_1U_2)=(TU_1)U_2$.

  • $TI = IT = T$

  • ParseError: KaTeX parse error: Double subscript at position 28: …U_1)U_2=U_1(a_U_̲2).

• These also holds for matrices.

• proof of matrix multiplication. In particular, the bracket is multiplicative.