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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