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- 10.Suppose that each of the vectors x(1), …, x(m) has n components, where n < m. Show that x(1), …, x(m) are linearly dependent. In each of Problems 11 and 12, determine whether the members of the given set of vectors are linearly independent for −∞ < t < ∞ . If they are linearly dependent, find the linear relation among them.arrow_forwardBased on other problems I believe the dimension is 3, but I'm not sure how to find a basis given this subspace.arrow_forwardIs S = {(x1, x2)^T ∈ R^2| x1 > x2} a subspace of R^2? Justify your answer.arrow_forward
- My question is verifying how V5 is a subspace of R^5arrow_forwardThe set of all points in R3 satisfying x + y - z = 0 is a subspace. Note that the set of point satisfying x + y - z = 2 is not a subspace (why?).arrow_forwardDetermine whether or not the following are subspaces of R^3. Justify your answers.1. {(x1, x2, x3)^T| x1 = x2 = |x3|}. 2. {(x1, x2, x3)^T| x3 = x1 + x2}.arrow_forward
- From my linear algebra course practice problems: "Is the subset of polynomials for which p(-1) = p(0) = p(5) a subspace of the vector space of all polynomials? Justify your reasoning." I'm aware of the 3 criteria needed to be met for a subset to be considered a subspace, but I just have no idea how to approach determining whether the zero vector is in the subset, and whether addition and multiplication are closed to the set.arrow_forwardFor Problem #12, how do I prove that the set is a basis for V? I think that infinity is the basis, but I'm not sure. This is a Linear Algebra type of question. Here is a picture.arrow_forwardFor question 6: Determine if the vector u is in the column space of matrix A and whether it is in the null space of Aarrow_forward
- The set of all vectors in R squared that satisfy 3x+4y=1 is a subspace of R squared . True Falsearrow_forward1-Suppose that S1and S2are nonzero subspaces, with S1 contained inside S2, and suppose that dim(S2)=3(a) What are the possible dimensions of S1? (b) If S1≠S2then what are the possible dimensions of S1? 2-Find the dimensions of the following linear spaces. (a) ℝ4×2(b) P3(c) The space of all diagonal 6×6 3-Find a basis {p(x),q(x)} for the vector space {f(x)∈P2[x]∣f′(4)=f(1)}where P2[x] is the vector space of polynomials in xx with degree at most 2. You can enter polynomials using notation e.g., 5+3xx for 5+3x^2p(x) , q(x)= 4-A square matrix is half-magic if the sum of the numbers in each row and column is the same. Find a basis BB for the vector space of 2×2 half-magic squares. B=arrow_forwardFind the Kernel and Range of the linear transformation L : R^3 → R^3 defined by L(x) = (x3, 0, x1)^T, where x = (x1, x2, x3)^T. What are the dimensions of these subspaces?arrow_forward
- Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning