7. Let T: V W be a linear transformation, and let V' be a subspace of...

 

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7. Let T: V W be a linear transformation, and let V' be a subspace of V. The restriction of T to V' is the function Ty: VW defined by Tv(v) =T(v) for all v V'. Prove that the restriction Tv is a linear transformation. 8. Let V and W be vector spaces and let T: VW be a linear transforma- tion. Prove that if (vv) is a linearly dependent set of vectors in V, then (T(vi),T(v)) is a linearly dependent set of vectors in W. 9. Let V and W be vector spaces and let (V,W) be the set of linear transformations from V to W. Let T and 72 be linear transformations in (VW) and let e be a scalar. Define addition and scalar multiplication of linear transformations as follows: (T+72)(v) = T(v)+7(v) and (CT)(v)=cT(v) for all vectors v V. Prove that (V,W) is a vector space. 10. A polynomial with real coefficients is an expression of the form Let Rr] be the vector space of polynomials with real coefficients. Define the function D: RR] by differentiation: D(f)=D(ant"+an-11" 1++ azt + ail + ao) =na+ -1)an-2-2 ....+2ayt+as a. Prove that differentiation is a linear operator: Iff and g are polynomials, and ife R, then D(f+g) D(f)+D(g) and D(ef)=cD(f). b. Compute kernel(D) and image(D). c. If V is a finite-dimensional vector space and D: VV is a linear operator. then kernel(D) (0) if and only if image(D) V. Use this to prove that the vector space Rr] is infinite-dimensional. Construct a basis for the vector space Rir]. 11. Let R] be the vector space of polynomials with real coefficients and let D: R] R] be the differentiation function. Compute the kernel of D for all positive integers k. 7. Let T: V W be a linear transformation, and let V' be a subspace of V. The restriction of T to V' is the function Ty: VW defined by Tv(v) =T(v) for all v V'. Prove that the restriction Tv is a linear transformation. 8. Let V and W be vector spaces and let T: VW be a linear transforma- tion. Prove that if (vv) is a linearly dependent set of vectors in V, then (T(vi),T(v)) is a linearly dependent set of vectors in W. 9. Let V and W be vector spaces and let (V,W) be the set of linear transformations from V to W. Let T and 72 be linear transformations in (VW) and let e be a scalar. Define addition and scalar multiplication of linear transformations as follows: (T+72)(v) = T(v)+7(v) and (CT)(v)=cT(v) for all vectors v V. Prove that (V,W) is a vector space. 10. A polynomial with real coefficients is an expression of the form Let Rr] be the vector space of polynomials with real coefficients. Define the function D: RR] by differentiation: D(f)=D(ant"+an-11" 1++ azt + ail + ao) =na+ -1)an-2-2 ....+2ayt+as a. Prove that differentiation is a linear operator: Iff and g are polynomials, and ife R, then D(f+g) D(f)+D(g) and D(ef)=cD(f). b. Compute kernel(D) and image(D). c. If V is a finite-dimensional vector space and D: VV is a linear operator. then kernel(D) (0) if and only if image(D) V. Use this to prove that the vector space Rr] is infinite-dimensional. Construct a basis for the vector space Rir]. 11. Let R] be the vector space of polynomials with real coefficients and let D: R] R] be the differentiation function. Compute the kernel of D for all positive integers k.

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To prove that the restriction TV of a linear transformation T is also a linear transformation you ne... View the full answer