Question

Long Answer Question LetV = M2x2, the vector space of 2 x 2 matrices with usual addition and scalar multiplication. Consider the set S = {M1, M2, M3} where M [ {].m=[5_1], 25 = [3 1] 1. (6 marks) Determine whether Sis linearly dependent/independent. 2. (2 marks) What is the dimension of Span(S)? 3. (2 marks) Is S a basis for V? 4. (2 marks) Is S a basis for the space of 2 x 2 upper triangular matrices? Please write your answers neatly on one sheet of paper; show your work for part (1). After you submit this test and exit Lockdown Browser, submit a.pdf copy of your work through the Term Test 2 Long Answer Dropbox link located under the link for this test. The submission of the file does NOT need Lockdown.

Answer 1

# V= Maxa, the vector space of 222 matrices with usual addition and sealar multiplication. set saf Mi, M₂, May M, [], M2 - [...].M = [3] xet us consider the relation C. M, + C₂ M₂ +Cg M₂ - L 6:] Cu C2, C3 are Scalars. 0 C + c2+363 2 e, +302 - 2 C3=0 erez tez lu from aitez ac, + 3 c, +363 - 203 20 and , & t3+3 e3 ie 50 tez C3 = -59 2 4 +4 03 20 9 20 Henee Caso C2=0, e320 siplinearing independent. :. The dimension of Span (s) in (since sia linearly independent) 2. v= Max 2 ViR A The dimension of and span (S) is a 3 dimensional subspace of r sib not a basis for v 4. rangular also Now all the three matrices of s are upper sis lineaking independent sel- of three upper triangular matrices Again dimension, all 2XL upper triangular matrieed Hence s formo a basis for the space of all 2X2 upper triangular matrices.