Question

5. Let T:R3 → R3 be a linear transformation given by (C:) - 3 - () - ) - (1:1) - 11

(a) Determine the matrix of T.

(b) Determine if T is one-to-one.

(c) Determine if there is any vector ~v such that T(~v) = [ 1 1 2 ]

(d) Determine if T is onto.

Answer 1

Let TR R3 be a linear transformation (138) (C): 0-(0:1). 6 solution. To find (determine matrix of T. by mabux representation of t. Let A be matus representation of linear transformation T with respect to standard basis. of R² Then T(x) = Ax by definition. Now we determine A as follows, rii o rol 117 ro] A-1 I I - 0 1 TT 10 1 2 4 -2 JAN It follows that we have 12 4 - 2 10 AO = 1 3 3 [4 2 2 7ou1 Let us find inverse mahux by using augmented matrix.

R₂+R, - as II I olioa + 10 10 - 0 41ool R3 - TT - TO - 1 R₂ AR riolloo R-R₂ 0 o 1 1o Ratky Lo 1-10 riolloo - RR ₂ - Rz 0102 Lo A oi rio 110 o 1] RI-R₃ Lo oil ili o ri 001 -.- 0 1 0 2 1 1 Too11 lo Thus we obtain inverse matrive at ri lo77 I t 47 -1 -1 - 2 1 1 1-10 + 110 Hence, we have will [2 4 -27 a - 47 A = 7 3 3 o 2.1 1 2 1 1 1 0 I 14 2 22.

4 o 2 d = 3 o ol 2 0 -2 using The relation T(x) = A.x the formula for T(x) is abtained follaws. T(x) = An (4 0 2T 3o o as TWT NUN L2 0 4x +az The) 32 2x-27 42+2 z a) Mature of T ,TA) = 3x 2x-2Z b) To determine t as one-to-one or not as follows. I A linear transformation is one-to-one ift the equation T(x)=0 has the only trivial solution (xco) and if t is standard matrix of T , then columns of A are linearly independent.

1 3 0 0 Here T(x) = d.x - 14 OR 1 2 0 2 1 multiplying a my x gives a linear combination of columns of A. Means that a, tit .... andn =0 where de la & columns are linearly independent la Ian/ proving T(x) = one to one. ① given T(w) = (il wen 42taz KIN que 32 2x-az 4x t2z = 1 3x=1 =)/x = 1² 24-22=2 =) 2 -22 =2 (d) T(x) is not onto function