Linear Questions Vector Space.docx

Linear Algebra 1. Let 𝑉be the vector space of all real polynomials of degree ≤ 𝑛 together with the zero polynomial. Determine whether 𝑊 is a subspace of 𝑉, where W consists of the zero polynomial and all polynomials. (i) With integral coefficients and of degree ≤ 𝑛. (ii) Degree of ≤ 3. (iii) With only even powers of 𝑥 and of degree ≤ 𝑛. 2. Express the vector (2, −5, 3) in 𝑅3 as a linear combination of the vectors (1, −3, 2), (2, −4, −1) and (1, −5, 7). 3. Show that each of the following sets of vectors generates 𝑅3 (i) {(1, 2, 3), (0, 1, 2), (0, 0, 1)} (ii) {(1, 1, 1), (0, 1, 1), (0, 1, −1)} 4. Determine whether the set 𝑆 = {(1, 1, 2), (1, 0, 1), (2, 1, 3)} spans 𝑅3 .