Linear Algebra

[kulgamer]

let T: C^n ---> C^n be given by T (X1 X2 ... Xn) = (0X1... Xn-1) show that T is a linear transformation and is T invertible, justify.

T(aX + bY) = aT(x) + bT(x)

T(aX + bY) = T(a(x1,x2, ..., xn) + b(y1, y2, ..., yn))
= T(ax1 + by1, ax2 + by2, ... , axn + byn)
= (0, ax1 + by1, ax2 + by2, ... , axn-1 + byn-1)

Now I don't know how to proceed from here to show that T is a linear transformation and if it is invertible.

 

[daon2]

What if n=2?

Let (x,y) and (x',y') be elements of C^2.

Then T(a(x,y) + b(x',y')) = T(ax+bx', ay+by') = (0, ax+bx') = a(0,x) + b(0,x') = aT(x,y) + bT(x',y')

Can you show for higher dimensions now?

T is NOT invertible. Find a way T can send a non-zero element of C^n to 0. If it is invertible, it ONLY sends 0 to 0.