Problem 1: consider the set of vectors in R^3 of the form: Material on basis and dimension Problem 1: Consider the set of vectors in R' of the form < a-2b,b-a,5b> Prove that this set is a subspace of R' by showing closure under addition and scalar multiplication Find a basis for the subspace. Is the vector w-8,5,15> in the subspace? If so, express w as a linear combination of the basis vectors for the subspace. Give the dimension of...
3t Let W be the set of all vectors of the form 5 +5 5s Show that W is a subspace of R* by finding vectors u and v such that W=Span{u,v). 5s Write the vectors in Was column vectors 31 5 4 5t = su + tv 5s 5s What does this imply about W? O A. W = Span(u,v} OB. W = Span{s.t O C. Ws+t OD. W=u+v
8. Let W be the set of all vectors in R3 of the form a(8, 9, 1), where a is a real number. A. Let b and c be arbitrary real numbers such that b(8, 9, 1) and c(8, 9, 1) are in W. Is b(8, 9, 1) + c(8, 9, 1) in W? B. Let k be a scalar and let b be an arbitrary real number such that b(8, 9, 1) is in W. Is k(b(8,9,1)) in W?...
The set of vectors {x1, x2} spans a subspace W of R’, where x1 = 4 2 5 and x2 ܕ ܩ ܟ 6 -7 (a) Use the Gram-Schmidt process to find an orthogonal basis for W. (b) Then normalize this new basis, so that it is an orthonormal basis. (c) Once you've found an orthonormal basis, demonstrate that it is indeed orthogonal after normalization. (d) For a bonus 2 points, calculate a third vector orthogonal to your basis and...
The set of vectors {x1, x2} spans a subspace W of R’, where x1 = 4 2 5 and x2 ܕ ܩ ܟ 6 -7 (a) Use the Gram-Schmidt process to find an orthogonal basis for W. (b) Then normalize this new basis, so that it is an orthonormal basis. (c) Once you've found an orthonormal basis, demonstrate that it is indeed orthogonal after normalization. (d) For a bonus 2 points, calculate a third vector orthogonal to your basis and...
hint: H3. Let W1 = {ax? + bx² + 25x + a : a, b e R}. (a) Prove that W is a subspace of P3(R). (b) Find a basis for W. (c) Find all pairs (a,b) of real numbers for which the subspace W2 = Span {x} + ax + 1, 3x + 1, x + x} satisfies dim(W. + W2) = 3 and dim(Win W2) = 1. H3. (a) Use Theorem 1.8.1. (b) Let p(x) = ax +...
EXPLAIN STEP BY STEP In Exercises 13 through 18 determine if the set of vectors S forms a subspace of the given vector space. Give reasons why S either is or is not a subspace. xn) in 13. S is the set of vectors of the form (x1, X2, ..., xn) in R”, with the x; real numbers and x2 = x4. 14. S is the set of vectors of the form (x1, X2, . R”, with the xị real...
a +3b 2b -2c 3a c 2. CLO 5 (5pts) W is the subspace of all vectors of the form in R4, where a, b and c are arbitrary real numbers. Find a basis for W
Linear Algebra: 1. 1.9 #6 For the following W = Span({(2,6,5,-4),(5,-2,7,1),(3,-8,2,6)}) a. Assemble the vectors into the rows of a matrix A, and find the rref R of A. b. Use R to find a basis for each subspace W, and find a basis for W as well. Both bases should consist of vectors with integer entries. c. State the dimensions of W and W and verify that the Dimension Theorem is true for the subspaces.
6. Let W be the set of all vectors of the form W {(a,b,c): a – 2b + 4z = 0} Is W a subspace of the vector space V = R3?