Question

3. In the following question, we are going to prove that ker(T) = { } if and only if T is one-to- one. (Writing prove is like writing a little essay, with some good logical connection between each sentence.) (a) Let T:V - W a linear transformation between two vector spaces. Suppose ker(T)={0}. Show that T is one-to-one. (Hint: proof by contradiction, by assuming both ker(T)=ð and T is not one-to-one. Now, apply definition of kernel and one-to-one, what is the problem? See Lecture 11, for an example of prove by contradiction.) (6) Let T :V - W a linear transformation between two vector space. Show that if ker(T) is not{0}, then T is not one-to-one. (Hint: “A implies B” is logically to “Not B implies Not A”) (c) Consider T:P3-R“, defined by T(1)=21, (x)=(2, 7(x2)= ez and 1(x)= e, for which, e1, ez, ez, e, are standard bases for R“. Compute ker(T), and using ker(T) determine if T is one-to-one or not.

Answer 1

LO be linear transformation between a T: V W two vector spaces. Let Ker(T) = {0} ſhext = fa<v: 760) = 8 To prove one-to-one one to one. If possible, let T is not Tis one-to-one = I x FY EV Tim) TLY) not such that | 2 T) - 11%) = 0 - T(0-4)=0 ET is Lineas 7 (2-3)€ Ket (T) ={7} photo = 2-4 = 0 sa=y •Whion is a Contradiction. obrotom rf Hence our assumption was wrong that I is not onttoone V T is one - to- one.

T: VW is a Linear Transformation: and ker(t) = f*: (a) z o} Let T is one-to tone. CLAIM kerit) = { 03 1e+ & Ekert and ato aldizden since atkert TEA) 0 torn T and also T60) = 0 T is Linear] T(2) = ETTO) a to but 2) I is not one to one. This snows that ker T = to 7 > Kert={0} one to one By contrapositively, we have Kert #28 = T is not one-to-one © T: P3 IRA and T(I) -e, TEA) = T(a) = ez T(x3) = 64 > vt ker T C P₃ Ut ker(t) = UE VE P₂ and Tlo=0 VEP3 = v=e,.lt gat ezat tega3 [since {1, 2, 3, ny is a basis of P3

& T10) = T(C.lt gat ezarrega) 9 TLV) = e, TLI) + ez TCX) + (z Tca) + CAT (13) o = a, e, tez tez ez + ca ea c=0=2 = 3 = 24 {e, ez, ez, eq} is a basis of 1R4 pe, ez, ez, egy is a ser = v=0 Tbyo] of linearly independent -> vectors. kestr) = 407 > Tis One - to one.

Hope it's helpful for you!

Thank you...