Vector space with vector in Rn, Summaries of Applied Mathematics

Download Vector space with vector in Rn and more Summaries Applied Mathematics in PDF only on Docsity!

VECTORS IN R®: LINEAR | COMBINATION AND LINEAR | INDEPENDENCE 5.1 INTRODUCTION We have already done vectors in 2-space and 3-space. Here, we will study vectors in n-space; i.e, real Euclidean n-space (vectors in R”). Vectors in Rris the extension of vectors in 2-space and in 3-space. 5.2 ORDERED n-TUPLES OF REAL NUMBERS A set (x,, X2,...x,) of m numbers arranged in the order is called an ordered n-tuple of reals. For example: (1) In two dimensional coordinate geometry any point (3, 7) is nothing but an ordered 2-tuple. : (2) (-2,0,4) is 3-tuple (3) (4, 6, 8, 9) is 4-tuple ~, -1250 * (4) Each row of the matrix, A = 123 il is an ordered 4-tuple. Each column of the matrix A is an ordered 2-tuple. 5.3 VECTORS IN R” Vectors in R” can be represented horizontally (row vector) and vertically (column vector). Itis denoted as ' X= (Kj, XyX,) OF X= ly, Xp %,] % Definitions of vectors in R” Vectors in R” is the collection of n-tuples of the form (x,, x,, ...x,). ie; a matrix 1 x n. This % can x also be of the form a matrix n x 1; |? Xn