Al. Let T1(x, y, z) = (1-y+z, 2:0 – y + 2z, 2y + 2). (a)....

Question

Question

Al. Let T1(x, y, z) = (1-y+z, 2:0 – y + 2z, 2y + 2). (a). Is T1 one-to-one?

(b). Is T onto?

math algebra

Answer 1

Answer #1

A1. T₁: R³ →R³ is defined by T₁(x,y,z) = (x-y+z, 2x-y+2z, 2y+z).

Hence T₁ (e₁) = T₁ (1,0,0) = (1,2,0), T₁ (e₂) = T₁ (0,1,0) = (-1,-1,2) and T₁ (e₃) = T₁ (0,0,1) = (1,2,1).

Hence the standard matrix of T₁ is A(say) = [T₁ (e₁), T₁ (e₂), T₁ (e₃)] =

The RREF of A is

This implies that the columns of A are linearly independent and span R³.

Therefore,

(a). T₁ is one-to-one.

(b) T₁ is onto.

Answer 2

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