Linear Algebra

222 Chapter Three. Maps Between SpacesBut if the left-multiplier’s nonzero entries are in the same row then that row ofthe result is a combination.3.6 Example⎛ ⎞ ⎛1 0 2⎜ ⎟ ⎜1 2 3⎞ ⎛ ⎞ ⎛1 0 0⎟ ⎜ ⎟ ⎜0 0 2⎞ ⎛⎟ ⎜1 2 3⎞⎟⎝0 0 0⎠⎝4 5 6⎠ = ( ⎝0 0 0⎠ + ⎝0 0 0⎠)⎝4 5 6⎠0 0 0 7 8 9 0 0 0 0 0 0 7 8 9⎛⎜1 2 3⎞ ⎛⎞14 16 18⎟ ⎜⎟= ⎝0 0 0⎠ + ⎝ 0 0 0⎠0 0 0 0 0 0⎛⎞15 18 21⎜⎟= ⎝ 0 0 0⎠0 0 0Right-multiplication acts in the same way, but with columns.These observations about simple matrices extend to arbitrary ones.3.7 Example Consider the columns of the product of two 2×2 matrices.( ) ( )g 1,1 g 1,2 h 1,1 h 1,2g 2,1 g 2,2 h 2,1 h 2,2=()g 1,1 h 1,1 + g 1,2 h 2,1 g 1,1 h 1,2 + g 1,2 h 2,2g 2,1 h 1,1 + g 2,2 h 2,1 g 2,1 h 1,2 + g 2,2 h 2,2Each column is the result of multiplying G by the corresponding column of H.G( ) ()h 1,1 g 1,1 h 1,1 + g 1,2 h 2,1=h 2,1 g 2,1 h 1,1 + g 2,2 h 2,1G( ) ()h 1,2 g 1,1 h 1,2 + g 1,2 h 2,2=h 2,2 g 2,1 h 1,2 + g 2,2 h 2,23.8 Lemma In a product of two matrices G and H, the columns of GH are formedby taking G times the columns of H⎛⎞ ⎛⎞..G · ⎜⎝⃗h 1 · · · ⃗h n⎟⎠ = ..⎜⎝G · ⃗h 1 · · · G · ⃗h n⎟⎠.and the rows of GH are formed by taking the rows of G times H· · · ⃗g r · · ·(ignoring the extra parentheses)..⎛ ⎞ ⎛⎞· · · ⃗g 1 · · · · · · ⃗g 1 · H · · ·⎜⎝ .⎟⎠ · H = ⎜⎝ .⎟⎠.· · · ⃗g r · H · · ·.Proof We will check that in a product of 2×2 matrices, the rows of the product