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Consider a Euclidean space R n and the 3 equations y 1 = x 1 3 + x 2 − 1 , y 2 = x 1 3 , y 3 = x 1 + x 2 + 2 . Now, a transformation T : R 2 ↦ R 3 is defined by : T ( x 1 , x 2 ) = ( x 1 3 + x 2 − 1 , x 1 + x 2 , x 1 + x 2 + 2 ) . Find T ( 2 , 8 ) .
Submit your answer as y 1 + y 2 − y 3 .
The answer is 11.
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2 solutions
The concept is very simple.The given space is 2 dimensional and we have to transform R^2 to R^3. The point mentioned is (2,8). Putting the values of x1=2 and x2=8. And calculate the value of y1, y2 ,y3. y1= 15, y2=8 y3=12 So the Answer is y1+y2-y3 = 15+8-12= 11
Relevant wiki: Euclidean N Space
Here , the transformation is from R 2 , which has 2 variables, x 1 , x 2 . And , we have to map it to a 3 Dimensional space R 3 , which has 3 variables y 1 , y 2 , y 3 . So , the point mentioned above is ( 2 , 8 ) . Just simply putting the values x 1 = 2 , x 2 = 8 at the place of x 1 , x 2 and calculating the value of y 1 , y 2 , y 3 , we get
y 1 = 2 3 + 8 − 1 = 1 5
y 2 = 2 3 = 8
y 3 = 2 + 8 + 2 = 1 2
So , The Answer is : y 1 + y 2 − y 3 = 1 5 + 8 − 1 2 = 1 1 .