Exercise_8 - reduced row echelon form, we have • 1 1 1 1...

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THE CHINESE UNIVERSITY OF HONG KONG Department of Mathematics MAT2310 Linear Algebra and Applications (Fall 2007) Solution of Exercise 8 Name: Student ID: Class: Find the kernel of the linear transformation T ( x ) = 1 1 1 1 2 3 x from R 3 to R 2 . What is dim(ker T )? Is T one-to-one? Explain. Solution: We have to solve the linear system T ( x ) = 1 1 1 1 2 3 x = 0 0 . Using the elementary row operations and transforming the augmented system to the
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Unformatted text preview: reduced row echelon form, we have • 1 1 1 1 2 3 ‚-→ • 1 0-1 0 1 2 ‚ . Hence x 1 x 2 x 3 = t-2 t t = t 1-2 1 , where t is an arbitrary constant. The kernel of T is the line spanned by 1-2 1 in R 3 . Clearly, dim(ker T ) = 1. Since ker T 6 = { R 3 } , T is not one-to-one....
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