Question

Show the below map is linear. What is its kernel? Image? Provide dimensions for both.

O: M(R) + R", (A) = Av.

Note that  

Answer 1

Q: Mn (R) → 1RM 9(A) = AV , VERm. let, Ag Az EMn (IR) P(Art A2 = (Ait A₂) 2 - Art All = PLA) + 9(A2)...O let, CER PleA) = (a) = C(A) = C9(A)... ☺ 0 and ② shows that, 9 is a linear. Kernel ai a42 I a. a ans ... an les ... an let, A= I EMM CIR ani ana ang .. ann . and v= T24 & I ERM 7 T Av= an ag, al ... am 222 ... asm I am and .. ann. I x ... n ! nh ai ut 042 t... f an an az utaz2 Xt... farnxn Ern any 4+ ana hot ...t ann an

Kernel of a 2 { AC Mn (R): 8(A) = AU = Orn) 9, x1 t 042 X t.intanxon lai ar ... an a, arr. .. Chen L. L a ixt a, X. t ..ddon lani ana ... Ann! an, at ane X 4.4 annanol We know a result, of v defined by constraints, If whspace w is the then dimw= dimv - number of indopendent equation (cmstraints). so, dim (Kernel of 4) = dim Mn(R) - n nan 'We know. soroo..o] colo..07 0 0 . 0 0 0 0. 0 > is a basis of MICIR)

Imo is the linear span of the vectors alt too..on solo.. 079 Too... o olloo.. o Löj ool, too..ol oroo... 07 Now or 912 913 ... ani p. 0,14 € 12 %, & ... of alinan as aga az ... azn = are at all t... tarth I ano ang ang in ann I V l anyut ane de of...t annan so Ime is the line ar span of the veears 100- I is linearly set . independent So, Imp= f 10 ) so, dim Imp = n. since, 8: Mr CR Rn By Rank - nullity a nxn² matris so o represents theorem,

dim ker q) + dim (7m 8) = n2 Now, dim kere) + dim (Imp) = n²nth = n2 , so also satisfied rank-nullity theorem