math111_logo Vector and Matrix


Exercise Take the standard basis vectors e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1) of R3. Compute the linear combination x1e1 + x2e2 + x3e3. Then generalize your result to other dimensions.

Answer x1e1 + x2e2 + x3e3 = x1(1, 0, 0) + x2(0, 1, 0) + x3(0, 0, 1) = (x1, 0, 0) + (0, x2, 0) + (0, 0, x3) = (x1, x2, x3). In general, for the standard basis e1 = (1, 0, ..., 0), e2 = (0, 1, ..., 0), ..., en = (0, 0, ..., 1) of Rn, we have

(x1, x2, ..., xn) = x1e1 + x2e2 + ... + xnen.