5. Let U and V be subspaces of R". Then UnV and U + V are defined as followsUnV:= {x ER" x EU and x € V}U + V := {x+y € R" | x € U and y € V}(a) Suppose U = {(s, t,0)|s, t ≤ R} and V = {(0, u, v)) | u € R, v ≤ R}. Draw pictures ofU, V, U + V and UnV.

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Answered: 5. Let U and V be subspaces of R". Then…

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter5: Inner Product Spaces

Section5.CR: Review Exercises

Problem 54CR: Let V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector...

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Let V be an two dimensional subspace of R4 spanned by (0,1,0,1) and (0,2,0,0). Write the vector u=(1,1,1,1) in the form u=v+w, where v is in V and w is orthogonal to every vector in V.
Determine whether the set R2 with the operations (x1,y1)+(x2,y2)=(x1x2,y1y2) and c(x1,y1)=(cx1,cy1) is a vector space. If it is, verify each vector space axiom; if it is not, state all vector space axioms that fail.
}) Let P3 be the vector space of all polynomials of degree 3 or less in the variable r. Let P1 (x) P2 (1) P3 (x) 2-z+ z? - r, 3z + 3z2-3r, 2+ z-z', - 2r3 6. PA (x) = 3-z+ 2z? and let C = {p1 (x), P2 (æ), P3 (2), P4 (x)}. choose a. Use coordinate representations with respect to the basis B {1, r, r-,r'} to determine whether the set C forms a basis for Pg. b. Find a basis for span(C). Enter a polynomial or a comma separated list of polynomials. { } c. The dimension of span(C) is S07 PM
3. Let W C R be an inner product subspace with standard dot product. = 0}, then I2 If W = I1 - 2x2 + r4 = I3 X4 (a) Find an orthagonal basis for W, (b) Find W.
21. Let V and W be subspaces of R² that are spanned by (3, 1) and (2, 1), respectively. Find a vector v in V and a vector w in W for which v +w = (3,5).
Let P3 is the vector space of polynomials of the degree<3,Consider the polynomials P1 t)=1+t,^p2(t)=1-t, is{p1, P2} a linearly independent set in P3? Why or why not.
Show that the set of all pairs of real numbers (x, y) with the operations (X1, Y1) + (x2, Y2) = (x1 + x2 + 1, y1 + Y2+ 1) and k (*1, Y1) = (kx1, kyı) is not a vector space.
X2 Let W be a subset of the vectors in R where W = X4 - X3 = x, – x }. Show that W is a X3 X4 subspace of R“. 3.
Let U1, U2, U3 – be the following subspaces of R4 Ui={(a, b, c, d)eR*| 2a - 3b + c + d=0, 0a - 3b - 2c - 2d=0}; U2= {a, b, c, d)eR| a=4b; c=d=0}; U3={a, b, c, d)eR“| a-b+c-d=0, 2a - 2b + c + d=0}; Sub-Task 1. Find a basis and the dimension for each Ui, i=1, 2, 3 Sub-Task 2. Whether R4 is the sum of U; and Uk, Hint. You must check 3 cases (i, k)=(1, 2), (i, k)=(1, 3), (i, k)=(2, 3) Sub-Task 3. If for some pair (i, k) R=U+Uk, check whether the sum is direct.

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5. Let U and V be subspaces of R". Then UnV and U + V are defined as follows UnV := {x € R" | x € U and x € V} U + V := {x+y ≤ R" | x ¤ U and y ¤ V} (a) Suppose U = {(s, t,0)| s, t € R} and V = {(0, u, v)) | u ≤ R, v ≤ R}. Draw pictures of U, V, U + V and UnV.