(a) Find an orthonormal basis for the subspace U = span ((1, −1, 0, 1, 1),(3, −3, 2, 5, 5),(5, 1, 3, 2, 8)) of R 5 .
(b) Express the vectors (0, −6, −1, 5, −1) as linear combinations of the orthonormal basis obtained in part (a).
(c) Which of the standard basis vectors lie in U?
(a) Find an orthonormal basis for the subspace U = span ((1, −1, 0, 1, 1),(3,...
ote: The norm of is denoted by |vand is calculated N a vector u Consider a subspace W of R4, W span(1, v2, v3, v4)). Where 0 из- 1. Find an orthonormal basis Qw of W and find the dimension of W 2. Find an orthonormal basis QWL of WL and find the dimension of WL 3. GIven a vector u- . find the Qw coordinate of Projw(v) . find the Qwa coordinate of Projwi (v) » find the coordinate...
(a) Find an orthonormal basis for the linear subspace V of R4 generated by the vectors 1 1 1 1 2 (b) What is the projection of the vector on the linear subspace V?
Q6. Let W be the subspace of R' spanned by the vectors u. = 3(1, -1,1,1), uz = 5(–1,1,1,1). (a) Check that {uj,uz) is an orthonormal set using the dot product on R. (Hence it forms an orthonormal basis for W.) (b) Let w = (-1,1,5,5) EW. Using the formula in the box above, express was a linear combination of u and u. (c) Let v = (-1,1,3,5) = R'. Find the orthogonal projection of v onto W.
3. Use the Gram-Schmidt process to find an orthonormal basis for the subspace of R' spanned by the vectors u; = (1,0,0,0), 12 = (1,1,0,0), uz = (0,1,1,1).
Find an orthonormal basis for the subspace of R3
spanned by
Extend the basis you found to an orthonormal basis for R 3 (by
adding a new vector or vectors). Is there a unique way to extend
the basis you found to an orthonormal basis of R3 ?
Explain.
Find an orthonormal basis for the given subspace. (Enter sqrt(n) for n.) S = span
Problem 2 [2 4 6 81 Let A 1 3 0 5; L1 1 6 3 a) Find a basis for the nullspace of A b) Find the basis for the rowspace of A c) Find the basis for the range of A that consists of column vectors of A d) For each column vector which is not a basis vector that you obtained in c), express it as a linear combination of the basis vectors for the range of...
5 5 8 form an orthogonal basis for W Find an The orthonormal basis of the subspace spanned by the vectors is (Use a comma to separate vectors as needed.) The vectors V, -2 and 12 - -3 3 orthonormal basis for W
Please attempt both questions.
5. Find an orthonormal basis for the plane viewed as a subspace of R3. Z (-1,0,2) (0,-1,0) (0,1,0) X 6. Determine if each basis is orthogonal. Further, is the basis orthonormal? (a) In the vector space R3 (i.e. column vectors in 3-space): 1 2 5 -3 (b) In the vector space that consists of polynomial functions of degree less than or equal to 2: {f(x) = 22 - 3, 9() = 4, h(x) = 2² +2}...
4. Use the Gram-Schmidt Process to find an orthonormal basis for the subspace of R5 defined by 2 S-span 0 2