Question

suppose T€ L(R^3) has an upper-triangular matrix with respect to the basis (1,0,0),(1,1,1),(1,1,2). Find an orthonormal basis of R^3 using the usual inner product on R^3 with respect to which T has an upper-triangular matrix.

Answer 1

Using Gram-schmidt principle

e_1=(1,0,0)\\ f_2=(1,1,1)-[(1,1,1)\cdot(1,0,0)](1,0,0)\\ =(0,1,1)\\ e_2=\frac{f_2}{\|f_2\|}\\ e_2=\frac{(0,1,1)}{\sqrt{2}}=(0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})\\ f_3=(1,1,2)-[(1,1,2)\cdot(1,0,0)](1,0,0)-\\ [(1,1,2)\cdot(0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})](0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})\\ =(1,1,2)-(1,0,0)-\frac{3}{\sqrt{2}}(0,\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})\\ =(0,-\frac{1}{2},\frac{1}{2})\\ e_3=\frac{f_3}{\|f_3\|}\\ e_3=\frac{ (0,-\frac{1}{2},\frac{1}{2}) }{\frac{1}{\sqrt{2}}}\\ e_3= (0,-\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})