Check that S is a linearly independent set over . (Hint: Consider the equation

.
 Put , etc. and solve for a_i. ) 

जाँच कीजिए कि f(x) सुपरिभाषित और 1-1 है। 

Check that  for  and  for . 

 मूल बिन्दु से समतल  के लंब की दिक्कोज्याएं ज्ञात कीजिए। 

 Find the inverse of the matrix B in part a) by using Cayley-Hamilton theorem. 

Let V be the set of all functions that are twice differentiable in  and

.

 Find the orthogonal canonical reduction of the quadratic form x² - 2y² + z² + 2xy + 6yz and its principal axes. Also, find the rank and signature of the quadratic form.