LinAlg4.1.4

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If $\bar{a},\bar{b}\,$ be two given vectors,show that the vector equation $\bar{x}+\bar{a}=\bar{b}\,$ has a unique solution $\bar{x}=\bar{b}-\bar{a}\,$

Let $\bar{c}=a\bar{a}+y\bar{b}\,$ where x and y are scalars.

$8\bar{i}+9\bar{k}=x(2\bar{i}+\bar{k})+y(3\bar{i}+4\bar{k})\,$

$(2x+3y)\bar{i}+(x+4y)\bar{k}\,$

$2x+3y=8,x+4y=9\,$

Solving these two equations,we get $x=1,y=2\,$

Therefore $\bar{c}\,$ as a linear combination of $\bar{a},\bar{b}\,$ is $\bar{c}=\bar{a}+\bar{b}\,$

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LinAlg4.1.4

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