Vellore Institute of Technology Department of Mathematics Digital Assignment for MAT3004 Ankush Chanda November 17, 2021 General Instructions: (Please read and follow strictly) 1. Clearly write your name, reg. no. and put your signature. If the student doesn’t record his/her name and registration number on each answer sheet, then his/her answer sheet will not be considered. 2. Rough sheet can be attached with the main answer sheet. 3. Take clear and visible snapshot of your filled-in answer sheet carefully and make a single pdf only and then upload it through log-in portal (VTOP). 4. Uploading of answers in any other format is not acceptable. Do not send different image files or zipped files. Do not send the answer sheet via e-mail. 5. The uploaded file will not be accepted after the due date, and the marks awarded will be automatically zero for those who do not submit in time. Do not postpone your task until the last date of submission. 6. If a student do not upload the answer script, then he/she will be treated as absent for the exam and will be awarded zero marks. 7. Use of any unfair means or any deviation from the above instructions will lead to the reduction in marks. Problems 1. Let S = {x1 , x2 , . . . , xk } be linearly independent vectors in a vector space. A. If we add a vector xk+1 to the collection, will we still have a linearly independent collection of vectors? Explain with an example. B. If we delete a vector, say xk , from the collection, will we still have a linearly independent collection of vectors? Explain with an example. 2. Let S be the subspace of P3 (R) consisting of all polynomials p(x) such that p(0) = 0, and let T be the subspace of all polynomials q(x) such that q(1) = 0. Find bases for S, T and S ∩ T. 3. Determine whether the following are linear transformations from C[0, 1] onto R: A. L(f ) = B. L(f ) = [f (0)+f (1)] 2 o 12 [f (x)] dx 0 nR 1 2 4. Find the standard matrix representation for each of the following linear operators: A. L is the linear operator that reflects each vector x in about the x1 axis and then rotates it 90◦ in the counter-clockwise direction. B. L is the linear operator that rotates each x in by 45◦ in the clockwise direction. 5. Let W be the subspace of R4 spanned by w1 = (2, 0, 3, −4), w2 = (4, 2, −5, 1), w3 = (2, −2, 14, −13), w4 = (6, 2, −2, −3). Is W = R4 ? If not, find a basis of W and extend it to a basis of R4 .