In Exercises 1 − 10 , the linear transformations S , T , H are defined as follows: S : P 3 → P 4 is defined by S ( p ) = p ′ ( 0 ) . T : P 3 → P 4 is defined by T ( p ) = ( x + 2 ) p ( x ) . H : P 4 → P 3 is defined by H ( p ) = p ′ ( x ) + p ( 0 ) . Also, B = { 1 , x , x 2 , x 3 } is the natural basis for P 3 and C = { 1 , x , x 2 , x 3 , x 4 } is the natural basis for P 4 . Find the matrix for T with respect to B and C .
In Exercises 1 − 10 , the linear transformations S , T , H are defined as follows: S : P 3 → P 4 is defined by S ( p ) = p ′ ( 0 ) . T : P 3 → P 4 is defined by T ( p ) = ( x + 2 ) p ( x ) . H : P 4 → P 3 is defined by H ( p ) = p ′ ( x ) + p ( 0 ) . Also, B = { 1 , x , x 2 , x 3 } is the natural basis for P 3 and C = { 1 , x , x 2 , x 3 , x 4 } is the natural basis for P 4 . Find the matrix for T with respect to B and C .
Solution Summary: The author explains the matrix for T with respect to B and C.
If {~v1,··· ,~vr} is linearly independent and T is a one to one linear transformation, show that {T~v1,··· ,T~vr} is also linearly independent. Give an example which shows that if T is only linear, it can happen that, although {~v1,··· ,~vr} is linearly independent, {T~v1,··· ,T~vr} is not. In fact, show that it can happen that each of the T~vj equals 0.
Consider the linear transformation T : R2[x] → R2[x] given by T(a + bx + cx2 ) = (a − b − 2c) + (b + 2c)x + (b + 2c)x2
1) Is T cyclic?
2) Is T irreducible?
3) Is T indecomposable?
Let L : P1 → P1 be a linear transformation defined by L(t − 1) = t + 2 and L(t + 1) = 2t + 1. (a) What is L(5t + 1)? (b) What is L(at + b)?
Chapter 5 Solutions
Introduction to Linear Algebra (Classic Version) (5th Edition) (Pearson Modern Classics for Advanced Mathematics Series)
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