I_l Question 1. Linear maps and linear independence {4+2+4 = 1!] marks) Let U and V be vector spaces, and let T : U } V be a linear transformation. Let the set ofp vectors {n1,.. . ,np} be s subset of vectors in U. 1. Show that if the set of vectors {u1, . . . ,up} is linearly dependent in U, then the set of iinages {T[u1}, . . . ,T(n,,}} is linearly dependent in V. The point of this is the following: If a linear transformation gives rise to a linearly independent set [the 'VEJCIIOI'S {T[u1}, . . . ,T(n,,)} are linearly independent in V}, then you know that the set of vectors they came from, the set {n1,. . . ,up} must be linearly independent in U. However, there is s catch. A linear map does not always transform a linearly independent set into a linearly independent set. 2. Gite an Example of a linear transformation that, when acting on a linearly in- dependent set of vectors {u1,...,np} in U, giyes rise to a linearly dependent set of vectors {T(n1},. . . ,T[np)} in V. Hint: think about the denition of linear independence. 3. Finally, suppose the linear transformation T is a onetoone transformation. That means, if T(n1} = T[ng), then 111 2 112. Show that if the set {T{u1}, . . . ,T(n,,}} of irnages under T is linearly dependent in F, then the set of vectors {n1,. . . ,up} is linearly dependent in U. Showing this proves that a onetoone transformation stands a linearly independent set to a linearly independent set because the set of images cannot possibly be linearly dependent. Question 2. Calculus: Application of sequences and series (5 marks) The Sierpinski carpet is a twodimensional counterpart of the Cantor set. It is constructed by removing the scuter oneninth of a square of side 1, then removing the cemers of the eight smaller remaining squares, and so so. (The gure shows the rst three steps of the construction.) Find that the sum of the areas of the removed squares and the area of the Sierpinski carpet