Page 1 of 1
Show that the euclidean metric d on R" is a metric, as follows: If x, y e R" and CER, define x+y= (x1+ yi, ..., Xn + yn)
Posted: Wed May 11, 2022 9:27 pm
by answerhappygod
- Show That The Euclidean Metric D On R Is A Metric As Follows If X Y E R And Cer Define X Y X1 Yi Xn Yn 1 (74.81 KiB) Viewed 25 times
Show that the euclidean metric d on R" is a metric, as follows: If x, y e R" and CER, define x+y= (x1+ yi, ..., Xn + yn), cx = (cx1, ..., cxn), X. = x.y = xiyi + ... + xn Yn. = (a) Show that x. (y + z) = (x. y) + (xz). (b) Show that x.y = ||* |||1y||. [Hint: If x, y = 0, let a = 1/||X || and b = 1/lly ||, and use the fact that ||ax + by|| > 0.] (c) Show that || x + yll = ||||| + ||y|l. (Hint: Compute (x + y) ยท (x + y) and apply (b).] (d) Verify that d is a metric.