9/16/2008

Inverse of a Linear Transformation

• Invertibility and rref(A)
• Finding the inverse of a matrix

Proof of the general form of the inverse for a 2x2 matrix

• Determinant of a 2x2 matrix

Matrix Multiplication

• To find BA, we can multiply B with the columns of A and combine the resulting vectors, as long as the dimensions of A and B permit multiplication.
• Matrix multiplication is noncommutative
• Multiplying with the inverse
• Multiplying with the identity matrix
• Matrix multiplication is associative
  • (AB)C = A(BC)
• The inverse of a product of matrices
• A criterion for invertibility
• Distributive property for matrices
  • A(C+D) = AC + AD
  • (A+B)C = AC + BC
• If k is a scalar, then (kA)B = A(kB) = k(AB)
• Partitioned matrices

Please "review" Appendix A

• Vector additon
• Scalar multiplication
• Rules for vector algebra
• Dot product
  • v.w = v1 x w1 + v1 x w2 + ... + vn x wn
• Rules for dot products
• Length (or norm) of a vector
• Unit vector
• Perpendicular (or orthogonal) vectors
  • Two vectors, v and w, are called perpendicular if their dot product is equal to zero.
• Cross product

Please read Section 3.1 for class on Thursday