Exercise_7 - combination as the columns of a matrix: [ x ]...

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THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MAT2310 Linear Algebra and Applications (Fall 2007) Solution of Exercise 7 Name: Student ID: Class: Consider two basis T = { w 1 , w 2 } and S = { v 1 , v 2 } for a vector space V such that ( w 1 = 4 v 1 + v 2 , w 2 = - 6 v 1 + v 2 . (1) Suppose that x = 3 w 1 + w 2 (2) That is, suppose that [ x ] T = 3 1 . Find [ x ] S . Solution: Apply the coordinate mapping determined by S to x in Equation (2). Since the coordinate mapping is a linear transformation, we have [ x ] S = [3 w 1 + w 2 ] S = 3[ w 1 ] S + [ w 2 ] S . We can write this vector equation as a matrix equation, using the vectors in the linear
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Unformatted text preview: combination as the columns of a matrix: [ x ] S = £ [ w 1 ] S [ w 2 ] S / • 3 1 ‚ . (3) This formula gives [ x ] S , when we know the columns of the matrix. From Equation (1), we have [ w 1 ] S = • 4 1 ‚ and [ w 2 ] S = •-6 1 ‚ . Thus Equation (3) provides the solution: [ x ] S = • 4-6 1 1 ‚• 3 1 ‚ = • 6 4 ‚ . (4)...
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