# week05 - Math 43 Fall 2007 B Dodson Week 5 Spanning...

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Math 43, Fall 2007 B. Dodson Week 5: Spanning, Independence and Bases Is the vector b = [3 , 3 , 4] a linear combination of v 1 = [1 , - 1 , 2] , v 2 = [2 , 1 , 3]? Solution We have 3 3 4 = c 1 1 - 1 2 + c 2 2 1 3 = 1 2 - 1 1 2 3 c 1 c 2 , where we use matrix notation from 3.1. We can easily see the system of equations directly, without using this matrix product, but writing the vector equation as the matrix equation Ac = b gives the augmented matrix of the system directly as [ A | b ] . We will also say that b is in the span of the columns of A. Observe that this is the linear system with augmented matrix [ v 1 v 2 | b ] = 1 2 | 3 - 1 1 | 3 2 3 | 4 . Row reduction gives 1 2 | 3 0 3 | 6 0 - 1 | - 2 . We see that this is a consistent system. (why?) So YES, b is a linear combination. We are often asked to continue by finding an explicit linear combination, solving for c 1 , c 2 . For this, we continue to the reduced row echelon matrix. We find c 2 = 2 , c 1 = - 1 , and b = - v 1 + 2 v 2 .

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