Question

2. Consider the matrix 1 2 0 1 2 A= 2 4 1 3 3 -1 -2 3 2 -5 Someone kindly obtained the reduced echelon form for this ma- trix, and got 1 2 0 1 2 R = 0 0 1 1 0 0 0 0 0 (a) (5 pts) We can immediately conclude the dimensions of each of the four fundamental subspaces associated with this matrix. Do so, identifying explicitly which spaces you are talking about.

Answer 1

/ 1 2 0 1 1 1 0 2 1 21 a - A=124 1331 REL OO, I-I... 1-1-23.2 -55 / looo 0 0/.. 0 Rankca)2, So: Dimension of Column Space --- CCA) is si Dimension of Row Space (CAT) 152, Bu Ranki Nuity Theorem - Dimension of Nullspace_195-22.3. & Dimension of lift Null space is 03-2=1. so Dim CCA)-21 Dim RCAT) = 2, Dim NCAS 3 - Dim NCAT) – 1 ② The pivots are at column 1 & 3 so CCA: Span S 1012 Hul 1371 Row Specc. CCAP): IC! 20 12 II. COLO 11-177}. Nall from space at NCAS 7 Aseso R we get x + 2x₂ + xut 225 to - azt xy-2.5 = 0 Let x₂ = sayt, as v. 2 . 1 -2 51-21 17-21 io NCA): Tot, 22 7 NCA) 2 tz 24 011) toto . 5 LAN NUN

ri 2-1 2 4 - 2 Ihmoo I H2 laat t17 1 3 2 2 3 -5 Töö ol Ata o xit Plz= 0, het az=k 22+ 3x3 = 0 .... 2 L tot 7 MCAT) = Span 1-3 Now NCA) is I to CCAT). NCAT) is I to CCA). ② Column space is defined as - S Ax | xe CCAT) , so By defn Ax is in the coluhan space and I af Axão 2 either at NCA) or Reparo But xe Row Space & xzo. - -