Let A be a 2 times 2 matrix and V1 and V2 vectors in R2 Prov

Let A be a 2 times 2 matrix and V_1 and V_2 vectors in R^2. Prove that A(alpha V_1 + beta V_2) = alpha AV_1 + beta V_2.

Solution

we know that marrices follow the distributive law that says

A(b+c)= Ab + Ac

now if alpha and beta are some real scalars that is {alpha,beta E R - {0}}

and if we have tow vecotors V1 and V2 in R2 that is

V1 =x1 i +y1 j

and V2 = x2 i + y2 j

and if A is a 2X2 matrix then

we could use the distributive law would hold true on the expression

A(alpha*V1 + beta*V2) , --------------->(1)

hence (1) could be writen as alpha*AV1 + beta*AV2

hence A(alpha*V1 + beta*V2) = alpha*AV1 + beta*AV2
THis is possible as both the matrix and the vectors are in 2 dimensional space coordinate R2