Let U CR' be the subspace generated by (1, 1, 1,0, 1), (2, 1, 0,0, 1), and (0, 0, 1, 0, 0). LetV CR° be the subspace generated by (1, 1, 0, 0, 1), (3, 2, 0, 0, 2), and (0, 1, 1, 1, 1).(a) Determine a basis of U n V.(b) Determine the dimension of U + V.(c) Determine a basis of U + V.

The given problem is related with linear algebra. Given that U ⊂ ℝ5 is a subspace generated by 1, 1,…

Answered: Let U C R’ be the subspace generated by…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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The set = 211 {(2a, b, a - b): a, b = R} 0,00) quo & jon ai & Jon ai (o.A) toda wodewoled ol is a subspace of R³. (a) Show that B = {(2, 1, 0), (2, 2, -1)} is a basis for S. (b) Is S a point, a line, a plane or R³ itself? (c) Use Gram-Schmidt orthogonalisation to find an orthogonal basis for S that includes the vector (2, 1,0). ai deda odmun oduo ovidiecq to och
Find a basis for the subspace of R* consisting of all vectors of the form (a, b, c, d) where c = a + 7b and d = a- 6b. (A) {(1, 0, 1, 1), (0, 1, 7, —6)}; (В) {(0, 1, 7, 1), (0, 1, 1, —6)} (С) {(1, 1, 0, 1), (1, 0, 7, —6)} (D) {(0, 1, 7, 1), (1, 1, 0, –6)} (E) {(1, 0, 7, 1), (0, 1, 1, –6)} (F) {(1, 1, 0, 1), (0, 1, 7, –6)} (G) {(1, 0, 1, 1), (1, 0, 7, —6)} (Н) {(1, 0, 7, 1), (1, 0, 1, —6)}
If V is the subspace spanned by (1, 1, 0, 1) and (0, 0, 1, 0), find (a) a basis for the orthogonal complement V-L. (b) the projection matrix P onto V. (c) the vector in V closest to the vector b = (0, 1, 0, -1) in V¹.
Find a basis for the orthogonal complement of the subspace of Rª spanned by the following vectors. V1 = (1,-1,7,3), V2 = (2,-1,5,2), V3 = (1, 0, -2, -1) The required basis can be written in the form {(x, y, 1, 0), (z, w, 0, 1)}.
Let W be the subspace of R5 orthogonal to u1=(1, 3, -2, 1, 1) and u2=(0, 1, 1, -1, 0), u3=(0, 0, -2, 1, 2), that is, W=S⊥, where S={ u1=( 1, 3, -2, 1, 1), u2=(0, 1, 1, -1, 0), u3=(0, 0, -2, 1, 2)}. Find Subtask (1): a basis and the dimension of the subspace W. Subtask (2): an orthogonal basis of the subspace W.
2. Is the set of vectors of the form a subspace of R? Show your proof. La+2b] 3. Show that the vectors v = (0, 3, 1, -1); v, = (6, 0, 5, 1); v; = (4. -7, 1, 3) form a linearly dependent set in R*? 4. Express V1 in number 3 as linear combination of V2 and V3. 5. For which value of x do the following vectors form a linearly dependent set in R -1 (-1 -1 x,

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Let U C R’ be the subspace generated by (1, 1, 1, 0, 1), (2, 1, 0, 0, 1), and (0, 0, 1, 0,0). Let V CR° be the subspace generated by (1, 1, 0, 0, 1), (3, 2, 0, 0, 2), and (0, 1, 1, 1, 1). (a) Determine a basis of U n V. (b) Determine the dimension of U + V. (c) Determine a basis of U+V.