Answer to Question #42311 in Linear Algebra for Anurodh

Question

a) Which of the following functions are 1-1 and which are onto? Justify your answer.
i) f : R→R≥0 given by f(x) = x2 whereR≥0 is the set{x∈R|x≥0}.
ii) f : R→Rgiven by f(x) = x2+x+1.

b) If we consider the expression 1/(2−3x) as a function on R, what will be its domain and range? Will it have an inverse? Justify your answer.

c) Let a=( 1/2√2, √3/2√2, 1/√2)and b=(1/√2,0, 1/√2).
i) Find the direct cosines of aandb.
ii) Find the angle between aand b.

d) Check that the vectors u=(3 5, 4 5,0),v=(−4 5, 3 5,0) and w= (0,0,1) are orthonormal. Further, write the vector a= (1,−1,2) as a linear combination of the vectors.