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Question

If A is an m x n matrix, define the function y = f(x) = Ax for all vectors x in Rn. Show that f satisfies f(k1u + k2v) = k1f(u) + k2f(v) for all vectors u and v of the correct size for pre-multiplication by A and any scalars k1 and k2 . This property is called linearity.

[Hint: Replace f by the matrix A and use marix properties to prove the result. You must state which matrix properties you are using.]

Why does this immediately imply that the function y = h(x) = ABx is linear (has the linearity property) for any matrices A and B where AB is defined?

Thanks for any help! :D

Answer 1

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The second part follows from the first since the first result holds for all matrices A.

For the first part, let A = (a_ij). Then the (i,j)-entry of f(k1u + k2v) is

(A(k1u + k2v))_ij

= ∑(k = 1 to n) a_ik(k1u + k2v)_k

(by definition of matrix multiplication)

= ∑(k = 1 to n) a_jk (k1u_k + k2v_k)

(by definition of matrix addition and

scalar multiplication)

= ∑(k = 1 to n) [k1(a_jku_k) + k2(a_jkv_k)]

= k1 ∑(k = 1 to n) a_jk u_k + k2 ∑(k = 1 to n) a_jk v_k

= k1 (Au)_ij + k2 (Av)_ij,

(by definition of matrix multiplication)

which is the (i,j)-entry of k1f(u) + k2f(v). Thus f(k1u + k2v) = k1f(u) + k2f(v).