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Final-Fall-2002-03-N-Nahlus - ERIC 4 I‘NTVERSW LIBRARY OF...

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Unformatted text preview: ERIC 4 \ I‘NTVERSW LIBRARY OF calm”. N. Nahlus Mathematics 218 Final Exam. Time: 211:5. Jan 25, 2003 1. (24 %) Prove (concisely) or Disprove (by a counter example) in an inner product space. (a) ll“ +V|l2 +llw—VI|Z=2|lMH2 +2llv'il2 (b) If llu llilvl|=<u,v > &u,v are non — zero then u &vare not orthogonal (c) Non~zero orthogonal vectors are linearly independent. m x n matrices have the same row space, then they have the same rank and nullifl. same row space, then they have the same null space. i l (d) If two (e) If two m x n matrices have the (i) If A is an orthogonal n x n matrix then detA= (g) (For any square matrix A), A and A2 have the same row space. // 2. (20 %) (a) Find the least sguares solutions of the system {x + y = 0 & x + y =1 & x + y = 4} ible) solution of a non—consistent system AX = b '3 (b) What do we mean precisely by a least sguare (best pass (c) What do we know about arbitrary symmetric n x n matrices regarding eigenvalues & diagonalization ? ctions on the interval [0, 3]. (d) Apply the Cauchy-Schwarz inequality on the continous fun subspace W (given an on basis of W). (e) Write the orthogonal projection formula for a vector a on a 3 6 3. (15%)LetA= i 2 o 0 (i) Find the eigen values of A an__d a basis for each eigen space of A. A , P and D. (ii) Show that A is diagonalizable and find the exact relation between {Do not calculate P“). linear transformation of vector spaces. If T(a1), T(a2), ..., T(a“) are 4. (9 %) Let T: V—>W be a linearly independent and dimV=n, show that {a,, a2, ..., an }is a basis of V. 0 0 5 5. (9 %) Let T: V—>W be a li . ') State the rank-nullity theorem for T, (ii) then use it to show that if T is onto and dimV=dimW=n, then T must be is injective. // 1 2 6. (9%)LetA: 2 4 0 o (i) Show that rank A=2 (ii) Does the system AX = B have a solution for everyB in R3 ? Justify. on basis of an inner product space V. 7. (9 0/o) Let {bh b2, ...,bn , a1, a2, ..., am } be an Let B= span {bb b2, ..., bu} & A==span {21], a2, ..., ar11 }. Show that (i) V=AEBB (ii) B=Ai. 1 6 3 8 O 2 2 2 l 5 2 7 8. (5 %) For any symmetric n x n matrix A, show that A and A5 have the same null space. ...
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