16. Systems of Linear Equations 1 Matrices and Systems of Linear ...

March 31, 2013 16-8We can write the systemas a single vector equationa 11 x 1 + a 12 x 2 + . . . + a 1n x n = b 1a 21 x 1 + a 22 x 2 + . . . + a 2n x n.= b 2(3)a m1 x 1 + a m2 x 2 + . . . + a mn x n = b mAx = b (4)where A in the m × n matrix (a ij ), x is an unknown n−vector, and b isa known m−vector.Comment: Strictly speaking, the left and right sides of (4) are m × 1matrices. It is customary to ignore this and simply refer to these as m −vectors. Sometimes, we refer to 1 × n matrices as n − vectors. The contextwill make it clear what is being done.We wish to determine whether the system (3) has a solution, and, if so,how many solutions does it have. Also, we seek a convenient way to representthe solutions.Before we consider this in detail, we note that there is a nice geometricdescription of the expression (4).4.1 Linear Maps and MatricesConsider the map T (x) = A · x. This defines a map from R n to R m .To solve the equation (4), where A and b are given, we are looking fora vector x ∈ R n such that T (x) = b. That is, we want b to be in theset-theoretic image of T .The map T has very special properties. It is what is called a linear mapor a linear transformation from R n to R m .The definition is the following.Definition 4.1 The map T : R n → R m is called linear if it preserves theoperations of vector addition and scalar multiplication. That is, for any twovectors x, y ∈ R n and any real number α, we haveT (x + y) = T (x) + T (y),