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:

[answerhappygod]

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 1 (48.48 KiB) Viewed 69 times

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 Ty is a linear transformation. 8. Let V and W be vector spaces and let T: VW be a linear transforma- tion. Prove that if (v₁.....V) is a linearly dependent set of vectors in V. then (T(v₁).....T(v₂)) 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 T₁ and 72 be linear transformations in L(VW) and let e be a scalar. Define addition and scalar multiplication of linear transformations as follows: (T₁+T₂)(v) T₁ (v)+7₂(v) and (cT₁)(v)=cT₁(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 f=an" +a+ +ar+ao. Let Rr] be the vector space of polynomials with real coefficients. Define the function D: R]→→R] by differentiation: D(f) = D(ant"+an-it + + a₂t² + a₁t+ao) =nant+(n-1)an-2"-2 +2ayt+a a. Prove that differentiation is a linear operator: Iff and g are polynomials, and ife R. then Df+8)= D(f)+D(g) and D(cf)=cD(f). b. Compute kernel(D) and image (D). c. If V is a finite-dimensional vector space and D: V-V is a linear operator. then kernel (D)(0) if and only if image(D)= V. Use this to prove that the vector space R is infinite-dimensional. Construct a basis for the vector space Ri]. 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.