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Unformatted text preview: Starting with Two Matrices Gilbert Strang, MIT 1. INTRODUCTION The first sections of this paper represent an imaginary lecture, very near the beginning of a linear algebra course. We chose two matrices A and C , on the principle that examples are amazingly powerful. The reader is requested to be exceptionally patient, suspending all prior experienceand suspending also any hunger for precision and proof. Please allow a partial understanding to be established first. I believe there is value in naming these matrices. The words difference matrix and sum matrix tell how they act. It is the action of matrices, when we form Ax and Cx and Sb , that makes linear algebra such a dynamic and beautiful subject. 2. A FIRST EXAMPLE In the future I will begin my linear algebra class with these three vectors a 1 , a 2 , a 3 : a 1 D 2 4 1 1 3 5 a 2 D 2 4 1 1 3 5 a 3 D 2 4 1 3 5 : The fundamental operation on vectors is to take linear combinations . Multiply these vectors a 1 , a 2 , a 3 by numbers x 1 , x 2 , x 3 and add. This produces the linear combination x 1 a 1 C x 2 a 2 C x 3 a 3 D b : x 1 2 4 1 1 3 5 C x 2 2 4 1 1 3 5 C x 3 2 4 1 3 5 D 2 4 x 1 x 2 x 1 x 3 x 2 3 5 D 2 4 b 1 b 2 b 3 3 5 : (1) (I am omitting words that would be spoken while writing that vector equation.) A key step is to rewrite (1) as a matrix equation: Put the vectors a 1 , a 2 , a 3 into the columns of a matrix A Put the multipliers x 1 , x 2 , x 3 into a vector x A D 2 4 a 1 a 2 a 3 3 5 D 2 4 1 1 1 1 1 3 5 x D 2 4 x 1 x 2 x 3 3 5 Then A times x is exactly x 1 a 1 C x 2 a 2 C x 3 a 3 . This is more than a definition of Ax , because the rewriting brings a crucial change in viewpoint. At first, the x s were multiplying the a s. Now, the matrix A is multiplying the vector x . The matrix acts on x , to give a combination of the columns of A : Ax D 2 4 1 1 1 1 1 3 5 2 4 x 1 x 2 x 3 3 5 D 2 4 x 1 x 2 x 1 x 3 x 2 3 5 D 2 4 b 1 b 2 b 3 3 5 : (2) When the x s are known, the matrix A takes their differences. We could imagine an unwritten x D , and put in x 1 x to complete the pattern. A is a difference matrix . 1 One more step to a new viewpoint. Suppose the x s are not known but the b s are known. Then Ax D b becomes an equation for x , not an equation for b . We start with the differences (the b s) and ask which x s have those differences. Linear algebra is always interested first in b D : Ax D is 2 4 x 1 x 2 x 1 x 3 x 2 3 5 D 2 4 3 5 : Then x 1 D x 2 D x 3 D (3) For this matrix, the only solution to Ax D is x D . That may seem automatic but its not. A key word in linear algebra (we are foreshadowing its importance) describes this situation. These column vectors a 1 , a 2 , a 3 are independent . The combination x 1 a 1 C x 2 a 2 C x 3 a 3 is Ax D only when all the x s are zero....
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This note was uploaded on 08/10/2008 for the course MATH 250 taught by Professor Chanillo during the Spring '08 term at Rutgers.
 Spring '08
 CHANILLO
 Algebra, Matrices

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