Question

5. Consider the matrix A-O , where a, b, c are real numbers. In this problem, we consider what the са со dimensions of subspaces associated to A could be; these answers may depend on the values of a, b, and c, and we want to see what the possible dimensions are. (a) Could the null space of A be 0-dimensional? 1-dimensional 2-dimensional? (b) Could the column space of A be 0-dimensional? 1-dimensional? 2-dimensional? 10 marks]

Answer 1

ca cb uhene atb,c one real number . In -his problem, aecodide, what忙 dime ndlons of Sobspaces anocteded to A Could be thee enkwers mou depend on the vahuel of arb iand c, ond we toot to see that- 叶he Pddible dime nlion) and ra) could the rol space of A o-dimedional 21-dincnulional? ca cb a b null space of A is o-dimensioned f aoo, b o a, b EIR except Boh abb be zero- CA nutl space 's 2-dimensional not possibi eto make nou space of fa any value of ab.

A be -dimentionad1-climend ioncp b) coold the column spacc 2-dime nsional 9 Sol' column space of s a, ca) 2) celomn space of A TS 1-dimendioca if either qand c aie non zero band c e nonzero 3) not it is not possible to make colu mo space of a s 2-dimersional