Homework 2.1 (Question 4)
To show that Tu,Tv and Tw are linearly independent, the scaler triple product must not be zero.

[Tu,Tv,Tw] = Tu.(Tv x Tw) =
expressing the vectors u,v and w in its component form…
Tuiei.(Tvjej x Twkek) = eijkuivjwk(Tei.(Tej x Tek))…
How do I continue from there sir???

If T is not singular, if Tu,Tv and Tw are also linearly dependent, then ∃α,β and γ all real such that αTu+βTv+γTw=o. But u,v and w are linearly independent. This means that αu+βv+γw≠o.
αTu+βTv+γTw= T(αu+βv+γw)=o.
This means that αu+βv+γw=o. This states that set of linearly independent vectors is linearly dependent! That is a contradiction!

Thanks

Good evening sir,

Just to remind you. You said, you were to going to give us a PDF to enable us do the homework i.e Gurtin 2.6.1, 2.8.5 etc…

Thank you sir.

Homework 2.1 (Question 4)

To show that Tu,Tv and Tw are linearly independent, the scaler triple product must not be zero.

[Tu,Tv,Tw] = Tu.(Tv x Tw) =

expressing the vectors u,v and w in its component form…

Tuiei.(Tvjej x Twkek) = eijkuivjwk(Tei.(Tej x Tek))…

How do I continue from there sir???

If T is not singular, if Tu,Tv and Tw are also linearly dependent, then ∃α,β and γ all real such that αTu+βTv+γTw=o. But u,v and w are linearly independent. This means that αu+βv+γw≠o.

αTu+βTv+γTw= T(αu+βv+γw)=o.

This means that αu+βv+γw=o. This states that set of linearly independent vectors is linearly dependent! That is a contradiction!