Define T : p 2 → ℝ 2 by T ( p ) = [ p ( 0 ) p ( 1 ) ] . For instance, if p ( t ) = 3 + 5 t + 7 t 2 , then T ( p ) = [ 3 15 ] . a. Show that T is a linear transformation. [ Hint: For arbitrary polynomials p , q in p 2 , compute T ( p + q ) and T ( c p ). b. Find a polynomial p in p 2 that spans the kernel of T , and describe the range of T .
Define T : p 2 → ℝ 2 by T ( p ) = [ p ( 0 ) p ( 1 ) ] . For instance, if p ( t ) = 3 + 5 t + 7 t 2 , then T ( p ) = [ 3 15 ] . a. Show that T is a linear transformation. [ Hint: For arbitrary polynomials p , q in p 2 , compute T ( p + q ) and T ( c p ). b. Find a polynomial p in p 2 that spans the kernel of T , and describe the range of T .
Differential Equations and Linear Algebra (4th Edition)
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