Linear Algebra

Section III. Computing Linear Maps 197gives the matrix representing the map.(cos θRep E2 ,E 2(t θ ) =sin θ)− sin θcos θThe advantage of this scheme is that by knowing how to represent the image ofjust the two basis vectors we get a formula for the image of any vector at all;here we rotate a vector by θ = π/6.( )3−2t π/6↦−→(√ ) ( )3/2 −1/21/2 √ 33/2 −2( )3.598≈−0.232(We are again using the fact that with respect to the standard basis, vectorsrepresent themselves.)1.10 Example In the definition of matrix-vector product the width of the matrixequals the height of the vector. Hence, the first product below is defined whilethe second is not.( ) ⎛ ⎞11 0 0 ⎜ ⎟ ⎝ 0⎠4 3 12( ) ( )1 0 0 14 3 1 0One reason that this product is not defined is the purely formal one that thedefinition requires that the sizes match and these sizes don’t match. Behindthe formality, though, is a sensible reason to leave it undefined: the three-widematrix represents a map with a three-dimensional domain while the two-tallvector represents a member of a two-dimensional space.Earlier we saw the operations of addition and scalar multiplication operationsof matrices and the dot product of vectors. Matrix-vector multiplication is a newoperation in the arithmetic of vectors and matrices. Nothing in Definition 1.5requires us to view it in terms of representations. We can get some insight byfocusing on how the entries combine.A good way to view matrix-vector product is as the dot products of the rowsof the matrix with the column vector.⎛ ⎞⎛⎞ c 1⎛⎞.c 2 ⎜⎝a i,1 a i,2 . . . a i,n⎟⎠ ⎜ ⎟⎝ . ⎠ = .⎜⎝a i,1 c 1 + a i,2 c 2 + . . . + a i,n c n⎟⎠.c .nLooked at in this row-by-row way, this new operation generalizes dot product.