the i-th standard basis vector, ei is the vector (0,0,...,1,...,0) where the 1 is in the i-th place (all other entries are 0). for example, in R^3:

e1 = (1,0,0) (sometimes called the "x unit vector" ori)

e2 = (0,1,0) (somtimes called the "y unit vector" orj).

e3 = (0,0,1) (sometimes called the "z unit vector" ork), notations vary from text to text.

in the "standard" basis, the vector (a1,a2,...,an) has coordinates (a1,a2,...,an) (it really is a convenient basis to use, because of this).

so in problem (e) T(a1e1+a2e2+...+anen) = a1e1 + a1e2 +...+ a1en,

because (a1,a2,...,an) = a1(1,0,...,0) + a2(0,1,0,...,0) +....+ an(0,...,0,1) =a1e1 + a2e2 +...+ anen.

since every basis vector ej except e1 has 0 in the first coordinate, what will the 2nd through n-th columns be?

what is T(e1) = T(1,0,...,0)?