Theorem 1.21 Let (X,T) be a topological space and Y₁, Y2,,Yk, be a family of path connected subspaces of X. If Yi Yi+1 Ø, Vi = 1, 2,... then Y₁UY2U... UYU... is path connected.
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- Find an orthonormal basis for the subspace of Euclidean 3 space below. W={(x1,x2,x3):x1+x2+x3=0}Give an example showing that the union of two subspaces of a vector space V is not necessarily a subspace of V.}) 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
- Show that W={(x1,x2,x3,x4) : x4-x3=x2-x1} is a subspace of R^4 spanned by vectors (1,0,0,-1),(0,1,0,1),(0,0,1,1)6. Prove that an open interval (a, b) considered as a subspace of the real line is homeomorphic to the real line.Let V and W be the subspaces of the vector space R^4 spanned by V1= (3,-1,4,1), V2 = (5,0,5,1), V3 = (5,-5,10,3) and W1 = (9,-3,3,2), W2 = (5, -1,2,1), W3= (6,0,4,1), respectively. Find the bases and dimensions for V + W and V ∩ W, and hence prove that dim(V + W) = dim(V) + dim(W) - dim(V ∩ W).
- 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.Let U1, U2, U3 – be the following subspaces of R+ Ui={a, b, c, d)eR*| 2b=c; a=d=0}; Uz={(a, b, c, d)eR*| a - b+ 3•c + 4rd=0, -a +b - 3c - 4 d=0}; U3= {a, b, c, d)eR*| 2a – 3:b – 2 c - d=0, a + 2b + Oc +,2 d=0 }; с, Sub-Task 1. Find a basis i and the dimension ( ) of U1. Sub-Task 2. Find a basis (v *) and the dimension ( f U2. Sub-Task 3. Find a basis .- and the dimension ( ;of U3. Sub-Task 4. Whether R=U1+U2, (provide a justification). ., '). Whether R=U1OU2, (provide a justification). (- :) Whether R=U¡©U3, Sub-Task 5. Whether R=U1+U3, (provide a justification (provide a justification). ( ). Sub-Task 6. Whether R=U2+U3, (provide a justification). (u. ). Whether Rª=U2©U3, (provide a justification). (*. ').Let (V,(*,*)) be an F-inner product space,where F is either R or C. LetU C V be a subspace and let a .= {u1,u2,...,uk} be an orthonormal basis for U. For each v E V, we defined proja(v) .= (v,u1)ui + (v,u2)u2+ ·… + (v,uk)uk. Prove that proja(v) is the closest vector to v in U and that it is the unique such vector, i.e. for all u EU, a) Iv – ul Iv – proja(v)l b) if Iv-ul=Iv-proja(v)l then u = proja(v)
- Verify that the set <v1, v2, . . . , vn> defined in Example 7 is a subspace.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.Let (V,(*,*)) be an F-inner product space,where F is either R or C. LetU C V be a subspace and let a .= {u1,u2,..,uk} be an orthonormal basis for U. For each vEV, we defined proja(v) .= (v,u1)ui + (v,u2)u2 + ·…· + (v,uk)uk. Prove that proja(v) is the closest vector to v in U and that it is the unique such vector, i.e. for all u EU, a) Iv- ul Iv- proja(v)l b) if Iv-ul=lv-proja(v)l then u = proja(v)