Question

This is a linear algebra question

(2) (a) Important theorem from linear algebra. The system of linear equations + ain^n = b1 a11 aml1 +amnTn = has either solutions (i) (ii) exactly (iii) Fill in each blank with a number, and show that this is true. Hint: Use the fact that every system of equations is equivalent to a system in echelon form. (b) Assume the above equations change the above theorem? (c) Assume further that the equations are homogeneous and that n > m. How does this change the above theorem? solution, or solutions are all homogeneous, i.e. bi = 0 for all i. How does this

Answer 1

2. a. The system of linear . aun att equasion ainan ab, .. . au 4 au 2 H, t . S . ami24 + . tamm uma bm Solution has rithes i infinitly manny (d) t exactly one (4) i No (o) solurion Solusion or The system can be whirrem as, 242 . um mi - Amm2 - I How er echolam fosm of A be ri 212 .. am 72 1 . .. Cum I ? oo.. . Lam J - Lom J then the gium system is equivalent to, 247 92 M2 +.. + Gimana bis una b'm . Naw just by substitution of an = b'm we ger The Bo solusion for an ein,.. nm. Il rank of Reasten posm of A be less than the number apuasiader' n then the system will have

1 lithes infinitly 4 the rank be the system will solusion of Mo Salarisan. equal to mam then , have exactly one solurion. 2 .b. H bico 4 d. then, the theorem changes to either som enactly have susrem have ame(') solurion, the P ( namely (0,0,0.. the bystem may have ) infinitly manny (d) Solution Il becote 4 ms m. then either Then, of 1 the system hare no (0) Solletion the system haue infinire (9 Soluhrs. in