Question

Hi, could you post solutions to the following questions. Thanks.

2. (a) Let V be a vector space on R. Give the definition of a subspace W of V 2% (b) For each of the following subsets of IR3 state whether they are subepaces of R3 or not by clearly explaining your answer. 2% 2% (c) Consider the map F : R2 → R3 defined by for any z = (zi,Z2) E R2. 3% 3% 3% 3% i. Show that F is a linear map. ii. Find the Kernel of F, Ker(F) and its dimension. ili. Find the Image (or Range) of F, Im(F) and its dimension. iv. Does the following sum dim (Ker(F)+ dim(Im(F) add up to the correct number? Explain your answer (a) Let V be a real vector space and let U, W be two subspaces of V. 3. i. Define the set U+W and show that it is a subspace of V. Spring 2017/s i. What does it mean that the sum (1) is direct? iii. State the Grassmann formula for U W and explain what are the conditions 1% 4% for the formula to hold true. What does the Grassmann formula simplify to if the sum (1) is direct. (b) Consider the two vectors in R th=(1,2) t12 = (4,1). ; [5% i. Show that (m) is a subspace of R3. Similarly, (a) is also 8 subspace of R. What are (u) and (u2)? Sketch a graph of them [5%] ii. Show that (R2 is the direct sum of (u) and (ua)) 5% then list their (a) Define the Euclide n inner product and the Euclidean norm on R 4 properties. (b) Consider the three vectors in R3 ut = (1,1,1); ua = (6,4,5); u,-(3,6,9). i. Show that {ul,u2,w) does not form an orthogonal set of vectors in Ra with i. Apply the Gram-Schmidt orthogonalisation process to (us ug, ts) to find an respect to the Euclidean inner product orthogonal set (v,s for R3 (with respect to the Euclidean inner product). Check that your answer is correct, them explain why (o,, ) is a basis for R3 4% ii. Use (r,,t) you found in i. to find an orthonormal basis for R 3% (c) Give the definition of an orthogonal matriz and show that if A is orthogonal, then its determinant is either 1 or -1. Provide an example of an orthogonal matrix and justify your answer Page 3 of 4 (a) Give the definition of row rank, column rank and rank of a matrix, then find the rank of each of the following matrices by clearly explaining your answer. 0 5 1 1-(0 0 o 3 Is C invertible? Explain your answer. 0-3-6C2 0 7 0 0 (b) Consider the two bases e and b of R2 given below e={(1,0),(0, 1)); b=((-1,1), (2,2)) i. Determine the matriz of change of coordinates from basis b to basis e, Mell). 1 i. Letv(-4,-4) be a vector in R2 (the coordinates of v are here given with 3 respect to the canonical basis e). Determine the coordinates of v with respect to the basis b. Check that the result you obtained is correct. (c) Find a matrix P that diagonalises 0 0-2 A 1 2 1 1 0 3

Answer 1

You have asked too many questions at one go. However, I have given answers to Q2, a, b (i), (ii), (iii), c) (i), (ii), (iii), (iv)

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