Appendices

Let x j ∈ ℜ n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315117218/6fa0f7e7-8893-4446-a58e-8202323d2be7/content/math19_1.tif"/> for j = 1 … m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315117218/6fa0f7e7-8893-4446-a58e-8202323d2be7/content/math19_2.tif"/> . As usual, x i j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315117218/6fa0f7e7-8893-4446-a58e-8202323d2be7/content/math19_3.tif"/> denotes the ith component of the vector x ^j . That is, x j = ( x 1 j , x 2 j , … , x n j ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315117218/6fa0f7e7-8893-4446-a58e-8202323d2be7/content/math19_4.tif"/> . A linear combination of x 1 … x m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315117218/6fa0f7e7-8893-4446-a58e-8202323d2be7/content/math19_5.tif"/> is a weighted sum of these vectors, ∑ j = 1 m α j x j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315117218/6fa0f7e7-8893-4446-a58e-8202323d2be7/content/math19_6.tif"/> where α j ∈ ℜ   ∀ j https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315117218/6fa0f7e7-8893-4446-a58e-8202323d2be7/content/math19_7.tif"/> . Geometrically, the linear combinations are all the points you can get to from the origin by taking steps of any lengths (including negative lengths) in any of the directions given by the vectors x 1 … x m https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315117218/6fa0f7e7-8893-4446-a58e-8202323d2be7/content/math19_8.tif"/> . In matrix notation, define the matrix X by X : , j ≡ x j , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315117218/6fa0f7e7-8893-4446-a58e-8202323d2be7/content/math19_9.tif"/>