Answer to Question #220154 in Linear Algebra for sabelo Bafana

Question

Suppose T is a normal operator on V. Suppose also that v, w $\in$ V satisfy the equations ||v||=||w||=2, Tv=3v, Tw=4w. Show that ||T(v+w)=10

Answer 1

given that ||v|| = 2 = ||w||

T(v) = 3v and T(w) = 4w

since v, w are eigenvectors, with distinct eigenvalue, therefore  v is orthogonal to w

$\implies$ 3v and 4w are also orthogonal

$\implies$$\langle$ T(v), T(w) $\rangle$ = $\langle$ 3v, 4w $\rangle$ = 0

now consider,

$\langle$ T(v + w), T(v + w) $\rangle$ = $\langle$ T(v) + T(w)), T(v)+ T(w) $\rangle$

= $\langle$ T(v), T(w) $\rangle$ +$\langle$ T(w), T(v) $\rangle$ +$\langle$ T(v), T(v) $\rangle$ +$\langle$ T(w), T(w) $\rangle$

= 0 + 0 + $\langle$ 3v, 3v $\rangle$ + $\langle$ 4w + 4w $\rangle$

= 3²$\langle$ v, v $\rangle$ + 4²$\langle$ w, w $\rangle$

= 3²||v||² + 4²||w||²

= 9$\cdot$ 2² + 16$\cdot$ 2²

= 36 + 64

= 100

$\implies$ ||T(v + w)||² = 100

$\implies$ ||T(v + w)|| = 10