3 Geometry of matrices Matrix algebra is extremely useful in the formulation and solution of sets of linear equations. At the same time, the algebraic results have a geometrical basis that is very helpful in understanding the calculation. It is helpful to digress to a geometric treatment of matrices and vectors 3.1 Vector Spaces 3.1.1 Euclidean Vector Space Perhaps the most elementary vector are the Euclidean vector space R", n 1, 2,.... For simplicity, let us consider first R2. Non zero vector in R2 can be represented geometrically by directed line segments. Given a nonzero vector we can associate it with the line segment in the plane from(0, 0) to (1, 12). If we equate line segment that have the same length and direction, x can be represented by any line segment from(a, b)to(a+a1, b+r2). For example, the vector x I in R2 could be represented by the directed line segment from(2,2)to(4,3), or fron(-1,-1)to(1,0) We can think of the euclidean length of a vector x= as the length of any directed line segment representing x. The length of the segment from(0, 0) Two basic operations are defined for vectors, scalar multiplication and ad dition. The geometric representation will help us to visualize how the operation of scalar multiplication and addition work in R2 (1). Scalar multiplication: For each vector x 1 d scalar a. the product ax is defined by a t1 For example, the set of all possible scalar multiple of x is the line through 0 and x. Any scalar multiple of x is a segment of this line Example3 Geometry of Matrices Matrix algebra is extremely useful in the formulation and solution of sets of linear equations. At the same time, the algebraic results have a geometrical basis that is very helpful in understanding the calculation. It is helpful to digress to a geometric treatment of matrices and vectors. 3.1 Vector Spaces 3.1.1 Euclidean Vector Space Perhaps the most elementary vector are the Euclidean vector space R n , n = 1, 2, .... For simplicity, let us consider first R 2 . Non zero vector in R 2 can be represented geometrically by directed line segments. Given a nonzero vector x =  x1 x2  , we can associate it with the line segment in the plane from (0, 0) to (x1, x2). If we equate line segment that have the same length and direction, x can be represented by any line segment from (a, b) to (a + x1, b + x2). For example, the vector x =  2 1  in R 2 could be represented by the directed line segment from (2, 2) to (4, 3), or from (−1, −1) to (1, 0). We can think of the Euclidean length of a vector x =  x1 x2  as the length of any directed line segment representing x. The length of the segment from (0, 0) to (x1, x2) is p x 2 1 + x 2 2 . Two basic operations are defined for vectors, scalar multiplication and ad￾dition. The geometric representation will help us to visualize how the operation of scalar multiplication and addition work in R 2 . (1). Scalar multiplication: For each vector x =  x1 x2  and each scalar α, the product αx is defined by αx =  αx1 αx2  . For example, the set of all possible scalar multiple of x is the line through 0 and x. Any scalar multiple of x is a segment of this line. Example: 6