Linear Algebra

194 Chapter Three. Maps Between Spaces1.4 Theorem Assume that V and W are vector spaces of dimensions n and mwith bases B and D, and that h: V → W is a linear map. If h is represented by⎛⎞h 1,1 h 1,2 . . . h 1,nh 2,1 h 2,2 . . . h 2,nRep B,D (h) =⎜⎟⎝ .⎠h m,1 h m,2 . . . h m,nand ⃗v ∈ V is represented by⎛ ⎞c 1c 2 Rep B (⃗v) =⎜ ⎟⎝ . ⎠c nBB,Dthen the representation of the image of ⃗v is this.⎛⎞h 1,1 c 1 + h 1,2 c 2 + · · · + h 1,n c nh 2,1 c 1 + h 2,2 c 2 + · · · + h 2,n c nRep D ( h(⃗v) ) =⎜⎟⎝.⎠h m,1 c 1 + h m,2 c 2 + · · · + h m,n c nDProof This formalizes Example 1.1. See Exercise 29.QED1.5 Definition The matrix-vector product of a m×n matrix and a n×1 vectoris this.⎛⎛⎜⎝⎞a 1,1 a 1,2 . . . a 1,na 2,1 a 2,2 . . . a 2,n⎟.⎠a m,1 a m,2 . . . a m,n⎛ ⎞1⎜⎝c.c n⎟⎠ =⎞a 1,1 c 1 + · · · + a 1,n c na 2,1 c 1 + · · · + a 2,n c n⎜⎝⎟. ⎠a m,1 c 1 + · · · + a m,n c nBriefly, application of a linear map is represented by the matrix-vectorproduct of the map’s representative and the vector’s representative.1.6 Remark In some sense Theorem 1.4 is not at all surprising because we chosethe matrix representative in Definition 1.2 precisely to make Theorem 1.4 true.If the theorem were not true then we would adjust the definition. Nonetheless,we need the verification that the definition is right.1.7 Example For the matrix from Example 1.3 we can calculate where that mapsends this vector.⎛⎜4⎞⎟⃗v = ⎝1⎠0