Question

IMPORTANT: If you are going to write please make sure your writing is neat, legible, and easy to read. Please write in print (not cursive). Thank you

IMPORTANT: The correct answers are shaded in. Please help me understand why the first, second, and fourth options are always true while the third, fifth, and sixth options aren't always true. Thank you

10. Which of the following statements are always true? Mark all that apply. If A is n x n and invertible, then Ax = b has a unique solution for every beR". If A is n x n and invertible, then its columns span R". If A is m x n and n <m, then the columns of A are linearly independent. If A is m x n and n > m, then the columns of A are linearly dependent. If A is m x n and n > m, then the columns of A span R". If A is m x n and rank(A) = n, then A is invertible.

Answer 1

d-10 Given År is Invertible We have ax=b multiply Bath sides with "A" À A k = AMG Ir = Ã In-ÅG ; tbe Rn/ By This Formula we can Always Find a Unique us" For All GERn rich That , An=6 Again briken " Ř is Invertible. column spon (A)=) An:RERA As we already brated in Port (a). If "A is Invertible , Then Fair All betet GERn There Enist x=Ã G such that Anak We hae For All GERn we hue An=6

es columns of "A" spun "Rna ii) Take Ang Exemple - of - 4 X 3 A=1 a s o Eis on m=4, n=3 But As one column is OOOO linearly so, colums of "A" are Nut Independent. so, This is not always true "A his "m" Rows and "n" columns and nom As we know a Runk IA) < min By Rank- Nullity. The crime Rank (A)t Null (A)=n Nallity (A) = n - RunkIA) > 0 Hence, 9 re Nall space (A); n&0 columns of À are linearly dependent

v we can see a counter Encmple Ti e a 4 o 0 here, m=2, n=3 As molt we have only one linearly Independent calumn, Hence It can not spon pm = R?? A = m = 3, n=2 Bath colums are linearly Independent. So, Runk (A) = n=2 A is Not Invertitle. But