We can think of the vector for a document as a point in |V |-dimensional space; thus the documents in Fig. 6 .3 are points in 4-dimensional space. Since 4-dimensional spaces are hard to visualize, Fig. 6 .4 shows a visualization in two dimensions; we've arbitrarily chosen the dimensions corresponding to the words battle and fool. Term-document matrices were originally defined as a means of finding similar documents for the task of document information retrieval. Two documents that are similar will tend to have similar words, and if two documents have similar words their column vectors will tend to be similar. The vectors for the comedies As You Like It [1, 114, 36, 20] and Twelfth Night [0,80,58,15] look a lot more like each other (more fools and wit than battles) than they look like Julius Caesar [7, 62, 1, 2] or Henry V [13, 89, 4, 3] . This is clear with the raw numbers; in the first dimension (battle) the comedies have low numbers and the others have high numbers, and we can see it visually in Fig. 6 .4; we'll see very shortly how to quantify this intuition more formally.