Linear Algebra and Its Applications (5th Edition)

General strategy for both parts: compute both the left hand side (LHS) and the right hand (RHS) side of the equation in component form and show that they are equal. (a) LHS: First, from the definition of vector addition: $u + v$ = ($u_{1}$ + $v_1$, $u_2$ + $v_2$, ... , $u_n$ + $v_n$) Therefore, from the definition of vector addition again: $(u + v) + w$ = ($u_{1}$ + $v_1$, $u_2$ + $v_2$, ... , $u_n$ + $v_n$) + ($w_1$, $w_2$, ..., $w_n$) = ($u_{1}$ + $v_1$ + $w_1$, $u_2$ + $v_2$ + $w_2$, ... , $u_n$ + $v_n$ + $w_n$) .......... (1) RHS: First, from the definition of vector addition: $v + w$ = ($v_{1}$ + $w_1$, $v_2$ + $w_2$, ... , $v_n$ + $w_n$) Therefore, from the definition of vector addition again: $u + (v + w)$ = ($u_{1}, u_2, u_3$) + ($v_1 +w_1, v_2 + w_2, ... , v_n + w_n$) = ($u_{1}$ + $v_1$ + $w_1$, $u_2$ + $v_2$ + $w_2$, ... , $u_n$ + $v_n$ + $w_n$) .......... (2) Since equation (1) is the same as equation (2) the property is verified. (b) LHS: First, from definition of vector addition: $u + v$ = ($u_{1} + v_1, u_2 + v_2, ... , u_n + v_n$) Second, from definition of scalar multiplication: $c(u + v) = (c(u_{1} + v_1), c(u_2 + v_2), ... , c(u_n + v_n))$ Third, from the distributive property of real numbers i.e. if $x, y, z$ are all real numbers then $z(x + y) = zx + zy$ we find: $c(u + v) = (cu_{1} + cv_1, cu_2 + cv_2, ... , cu_n + cv_n)$ .......... (3) RHS: First, from definition of scalar multiplication: $c\times u = (cu_1, cu_2, ... cu_n)$ $c\times v = (cv_1, cv_2, ... cv_n)$ Therefore, from the definition of vector addition: $c\times u + c\times v$ $= (cu_1, cu_2, ... cu_n) + (cv_1, cv_2, ..., cv_n)$ $= (cu_1 + cv_1, cu_2 + cv_2, ... , cu_n + cv_n)$ .......... (4) Since equation (3) is the same as equation (4) the property is verified.