Let W be a subspace of R". Recall (from Exercise 5 Q7) that w+ = {u € R" | u is orthogonal to W} is also a subspace of R". Show that W nW+ = {0}. Given a basis S = {w1, w2; · (i) ,wm} for W. Suppose we (ii) (I) extend S to a basis for R": ... T = {w1, w2, ·…' , Wm, Wm+1' wn} and (II) apply Gram Schmidt to T to get an orthogonal basis T' = {wi, w2, · wm, w, · ,w,}. ... +1;* wm} and W = span{wm+1; wm Show that W = span{w',, w, ·.. , Hint: Use Exercise 3 Q43. Also note that if U = span(X) and V = span(Y), then U +V = span(X UY). m+2;*** , w,}. Let A be an n x k matrix with W as the column space. Show (iii) that the solution space of AA"x = 0 is given by W+.

Q: Let V be a finite dimensional vector space with dimension n > 0. Let S = {V1, V2, Vn} be a family of…

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Q: 3. The set B = {1-t², t-t²,2-2t + t²} is a basis for P2. Find the coordinate vector of p(t) =…

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Q: 1. Let u = [1 0 1]",0₁ = [1 1+ T -i]¹, 1+i 1- - V₂ = [2i 1 0], and v3 V3 = [2 −1 1]². Let = V1, V2,…

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Q: 14. Let P₂ be the vector space of polynomials of degree ≤ 2. Let D = {1 + t, 1 − 5t, 2 + t²}. Check…

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Q: Consider the following vectors in V = R³ over F = R. v₁ = (1,0,1) U2 = (0,1,2) √3 = (1,2,0)

A: We have given a set of vectors.To check if the set is linearly independent and form a basis of V(F),

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Q: 3.) Consider the following vector. O M 2 2 t 2 6 x3 = 16 -S a) Are the vectors {x₁, X₁, X3) lineonly…

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Q: 4. s] Consider the following vectors in V = R³ over F = R. v₁ = (1,0,1) 2 = (0,1,2) √3 = (1,2,0) (a)…

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Q: The set B = {1+t², t + t², 1 + 2t + t²} is a basis for P₂. Find the coordinate vector of p(t) = 1 +…

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Q: Let the following vectors be in R³ a = (1,3,-1) b= (-2,1,3) c = (3,2,1) d = (1,-1,2) 16.- Obtain a…

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Q: 1 17. В %3D is an ordered basis of R². = , 2 find x. 3 (a) If the B-coordinate vector of a vector x…

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Q: 3. Given the two ordered bases U, V below for R³ (also the standard basis E = [e1, €2, €3]) (): 1…

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