Theorem 42 Let A be an m nmatrix A a11 a12 a 1 n a21 a 22 a 2 n a m1 am 2 amn

# Theorem 42 let a be an m nmatrix a a11 a12 a 1 n a21

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Theorem 4.2 Let A be an m × n matrix A = a 11 a 12 · · · a 1 n a 21 a 22 · · · a 2 n . . . . . . . . . . . . a m 1 a m 2 · · · a mn . and denote the columns of A by the column vectors c 1 , c 2 , . . . , c n , so that c i = a 1 i a 2 i . . . a mi , 1 i n. Then if x = ( x 1 , x 2 , . . . , x n ) T is any vector in R n , A x = x 1 c 1 + x 2 c 2 + · · · x n c n . This theorem states that the matrix product A x , which is a vector in R m , can be expressed as a linear combination of the column vectors of A . Activity 4.3 Prove this theorem; derive expressions for both the left-hand side and the right-hand side of the equality as a single m × 1 vector and compare the components to prove the equality. Read section 1.8 of the text A-H, working through the activities there. You will find the solution of the last activity in the text. 4.2 Developing geometric insight – R 2 and R 3 Vectors have a broader use beyond that of being special types of matrices. It is likely that you have some previous knowledge of vectors; for example, in describing the displacement of an object from one point to another in R 2 or in R 3 . Before we continue our study of linear algebra it is important to consolidate this background, for it provides valuable geometric insight into the definitions and uses of vectors in higher dimensions. Parts of this section may be review for you. 53
4 4. Vectors 4.2.1 Vectors in R 2 The set R can be represented as points along a horizontal line, called a real-number line . In order to represent pairs of real numbers, ( a 1 , a 2 ), we use a Cartesian plane , a plane with both a horizontal axis and a vertical axis, each axis being a copy of the real-number line, and we mark A = ( a 1 , a 2 ) as a point in this plane. We associate this point with the vector a = ( a 1 , a 2 ) T , as representing a displacement from the origin (the point (0 , 0)) to the point A . In this context, a is the position vector of the point A . This displacement is illustrated by an arrow, or directed line segment, with initial point at the origin and terminal point at A . - x 6 y (0 , 0) > r ( a 1 , a 2 ) a 2 a 1 a position vector, a Even if a displacement does not begin at the origin, two displacements of the same length and the same direction are considered to be equal. So, for example, the two arrows below represent the same vector, v = (1 , 2) T . - x 6 y (0 , 0) displacement vectors, v If an object is displaced from a point, say O , the origin, to a point P by the displacement p , and then displaced from P to Q , by the displacement v , then the total displacement is given by the vector from O to Q , which is the position vector q . So we would expect vectors to satisfy q = p + v , both geometrically (in the sense of a displacement) and algebraically (by the definition of vector addition). This is certainly true in general, as illustrated below. 54
4 4.2. Developing geometric insight – R 2 and R 3 - 6 (0 , 0) : > * p 2 p 1 q 2 q 1 p v q If v = ( v 1 , v 2 ) T , then q 1 = p 1 + v 1 and q 2 = v 2 + p 2 .