Question

mints) Give the following examples if possible. If it is not possible, explain why. You do not have ove the properties hold 2 points) A list of 4 vectors in Rồ that is linearly independent. (3 points) A subspace W of R* that is complementary to V = | 22 23=0, 1334=0 =) (4 points) A linear map T:R?R? with image(T) - ker(T) = span ()

Answer 1

IR is 3. Solution ① A list of 4 vectors in IR that is linearly independent Solution & It is not possible because of dimention of vector space Hence there exists at most 3 reeters sets which are lin linearly independent. Hence there does not exists a list of 4 vectors in R3 that is eine ineadly independent (2) А subspace k of IRA that is complementary to X2 X3=0, 214 = 0 au solution of kindly kindly note that kl = complement of va & to x420 23 24 0 & complement of v=W le. zero element TRY is not in complement of V. & W 0 0 Hence W is not subspace of IR L-e. A subspace kl of iR" that is complementary to v.

Now, [T] = 11 -12 2 perform R2 R₂ ZR, I -/ It is reduced now O O echelone form of [T] B & its first commn contain pirot element 1 Hence first column of CT] B forms Basis for Column space of [T] B be. forms column space of [T13 2 Henee forms Basis for Range spore of T (I) = span Range {] 0 T: R² - DR² x-9/2 2x-g T([%]) is required example