# Question

A function f: Rn x Rm→ Rp is bilinear if for x,x1, x2 € R n, y,y1, y2 € Rm and a € R\

We have,

f(ax, y) = af (x, y) = f(x, ay)

f(x1 + x2, y) = f(x1, y) + f(x2, y)

f(x, y1 +y2) = f(x, y1) + f(x, y2)

(a) Prove that if f is bilinear, then

(b) Prove that Df (a, b) (x, y) = f (a,y) + f(x,b).

(c) (Show that the formula for Dp (a, b) in theorem 2-3 is a special case of (b

We have,

f(ax, y) = af (x, y) = f(x, ay)

f(x1 + x2, y) = f(x1, y) + f(x2, y)

f(x, y1 +y2) = f(x, y1) + f(x, y2)

(a) Prove that if f is bilinear, then

(b) Prove that Df (a, b) (x, y) = f (a,y) + f(x,b).

(c) (Show that the formula for Dp (a, b) in theorem 2-3 is a special case of (b

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