6. A researcher collects expression measurements fer 1,000 genes in 100 tissue samples. The data can be written as a 1,000 x 100 matrix, which we call X, in which each row represents a gene and each col umn a tissue sample. Each tissue sample was processed on a different day, and the columns of X are ordered so that the samples that were processed earliest are on the left, and the samples that were processed later are on the right. The tissue samples belong to two groups: con trol (C) and treatment (T). The C and T samples were processed in a random order across the days. The researcher wishes to deter mine whether each gene's expression measurements differ between the treatment and control groups. As a preanalysis (before comparing T versus C), the researcher per forms a principal component analysis of the data, and nds that the rst principal component (a vector of length 100) has a strong linear trend from left to right, and explains 10 9513 of the variation. The re searcher now remembers that each patient sample was run on one of two machines, A and B, and machine A was used more often in the earlier times while B was used more often later. The researcher has a record of which sample was run on which machine. (a) Explain what it means that the rst principal component "ex plains 10 9% of the variation"