Prove as follows the inequality , where A is an m × m matrix with row vectors a1, a2, ....

Prove as follows the inequality , where A is an m × m matrix with row vectors a₁, a₂, . . . , a_m, and x is an m-dimensional vector. First note that the components of the vector Ax are a₁ ⋅ x, a₂ ⋅ x, . . . , a_m ⋅ x, so

Then use the Cauchy–Schwarz inequality for the dot product.