Question

Let?:R ⟶? (R)be definedas? $\begin{bmatrix} a\\ b\\ c\\ d \end{bmatrix}$ =(?−2?)+(?+3?)?+(?−2?)?² .

a. Find a basis for the Ker(T). (3pts)
b. Find a basis for the Range(T). (3pts)
c. Determine whether T is one-to-one. (2pts)

d. Determine whether T is onto. (2pts)

Answer 1

Here, TRAP defined as TOP (a-2d) & (e +36) x + (ar 2e) x2 Here; R = Rle p= P2 a ) Now Ker (1) - {XER: T(x)=0}, so 2 T(x) = 0; sout (ar20) + (9-36x + (a-2ex.o. a-2d 20. (+3620 a-20 Гоо 2 we write the systein an 0 b 0 3 1 0 o 0-2 ud - Apply now operation on the argumented mortrix to reduce it to vow reduced echelon form, so, 6 0-2 O O -2 0 0 1 o 0 -2 0 D 0-2 2 1 2 0 3 0 0 09 CBR - BAR 0 22 0 76 0 The System becomes, a-20 20 b + d zo. e - 2d zo.

a a zd batd, cold. 2d od X2 d YG +/ d So, Ker(T) Spen 2 -V6 V2 Borsis for the ker(i) is (6) Here, we can write, 0 0-2 T([]) 03 I J. 0.-2 AX. T(x) A is matrix of the linear foonsformation T. RREF (A) - 0 - 2 O VE 0 1-12 Now, Now, standard Bornis of Ris, {4, R2, les seu} 1,2 (1,0,0,0), ey? (0,1,0,0), ez? ( 0,0,1,0), Rey 2 (0,0,0,1) Range of Te Spom{(e), T@)7(e)), T(w)} Here, in the RREF (A), the ist three Columns Contains the leading is . S, T(R1), TCee), T(RB) form a basis of R(7). R (T) Range space of T.

1st three Colungs Basis for R (T) & is the of matrix Ai Basis for R (T) {fl-1:1 [:)] know that for o linear frons formation T, is one-one if and only if Ker(T) = {0}. But there, Kenia We a cheme, kuna) { Here, So, The lineon transformation I is out one-one. (d). P = P2 (P2 is the space of polynominks of degree less than or equal to 2 Basis for p in (1,0m) So, dim P and here, dim R (T) B. So, we can say that. I is onto. = 3.