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Vector sum $(x_1, x_2)+'(y_1, y_2)=(x_1+2y_1, 3x_2-2y_2)$ and the usual scalar multiplication $c(x_1, x_2)=(cx_1, cx_2)$.

Sure additive properties does not hold for the operation but why does the property $(c+d)x=cx+dx$ not hold either?

What did I miss?

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1 Answer 1

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Well you could try calculating the RHS of that expression, $$cx+dx=c(x_1,x_2)+d(x_1,x_2)=(cx_1,cx_2)+(dx_1,dx_2)=(cx_1+2dx_1,3cx_2-2dx_2),$$ compare it with the LHS which is $$(c+d)x=(c+d)(x_1,x_2)=\big((c+d)x_1,(c+d)x_2\big).$$ Are they the same?

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  • $\begingroup$ I presumed that we only use the scalar mulplication to test the property, but apparently I was wrong. Thanks for clarifying my confusion. $\endgroup$
    – CoolKid
    Commented Feb 1, 2016 at 18:08
  • $\begingroup$ By the way, is it true that if $x=(x_1,x_2)$ and $y=(y_1, y_2)$, $x\cdot y=(x_1y_1, x_2y_2)$? $\endgroup$
    – CoolKid
    Commented Feb 1, 2016 at 18:09
  • $\begingroup$ @CoolKid Remember, in a vector space $V$ over a field $F$ we usually work with two operations, addition between members of the vector space, $+\colon V\times V\to V$, and scalar multiplication between members of the field $F$ and members of the vector space, $\cdot : F\times V\to V$. Note that what you just did was to multiply two elements from the vector space using the definition of multiplication between elements of a field and those of a vector space. Do you feel that there is anything incorrect with that? $\endgroup$
    – Workaholic
    Commented Feb 1, 2016 at 18:16

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