Question: 7 (a) Let u = [3] E R2. Consider the linear transformation :R2> [R2 (ll-V V|> )u 11- u U where - denotes the dot

7 (a) Let u = [3] E R2. Consider the linear transformation :R2> [R2 (ll-V V|> )u 11-\" u U where - denotes the dot product on R2. (Recall: l I] - [ I] = \"101 + 14202.) \"2 U2 Geometrically, this linear transformation corresponds to a projection onto a line in R2 through the origin with direction vector u. Find a nonzero vector in the image of go. The vector W = [j is in the image of (p since w [j H ,5 (b) Consider the linear transformation l]! 2 P3 > M2(R) 5015b+lc+14d 3a+9b+648d axs +bx2 +cx+d I> [ 2a+6b+1c14d 1a3b+1c2d Find a nonzero vector in the image of w. The matrix A = [:[j is in the image of w since w( [:]x3+ [:Jx2+[:]x+[:] )zA. Check Decide whether each of the following maps is a linear transformation. Justify your answer in each case. (a) 41 : P S p - ((-1)"p(n2 - 1))n (b) 42 : P P -p if p(1) = 0 p p otherwise

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