*Problem 3.4 Suppose you start out with a basis (lei), le2), ..., len)) that is not orthonormal. The Gram-Schmidt procedure is a systematic ritual for generating from it an orthonormal basis (le), le2), ..., le)). It goes like this: (i) Normalize the first basis vector (divide by its norm): le) = le₁) lleill (ii) Find the projection of the second vector along the first, and subtract it off: le₂) - (ele₂) lei). This vector is orthogonal to le); normalize it to get le₂). (iii) Subtract from le3) its projections along le₁) and le₂): les) - (ejles) lej) - (e₂lez)|e₂). This is orthogonal to le) and le₂); normalize it to get leg). And so on. Use the Gram-Schmidt procedure to orthonormalize the three-space basis |e₁) = (1 + i)i + (1)ĵ+ (i)k, \e₂) = (i)î+ (3)ĵ+(1)k, |e3) = (0)î+ (28)ĵ+ (0)k.

3. Given a₁ = (1,1,1,1), a₂ = (1,-1,0,4), a3 = (3,5,1,-1), use the Gram-Schmidt process to obtain an orthonormal vector set.

9. Consider the orthonormal basis S for R' S = {v₁,v₂,v,} where v₁ = (0,1,0) S= v, -(-3.0,) and v, - (3.0.3). Let u = (1,0,4). Find (2), = 8 19 A (u), -(0,-3,1) = 8 19 B (u), - (1,3,7) = 8 19 C (u), = (0,3, 1/2) D No correct answer E (u) = (0,8,19)

(i) Give an example of two orthonormal vectors in R³. (ii) Using C=9 D=3 E=5 F=4 Schmidt process to find an orthonormal basis from the initial vectors () and (7). use the Gram- (iii) Construct a matrix which implements a shear of factor 3 along the first vector of your basis constructed in (ii), in that basis. Then use change of basis methods to give the corresponding (similar) matrix in the standard basis of R². Homework