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MTM 5245 - A´lgebra Linear - Gabarito Lista de Exercı´cios 04 Base, dimensa˜o, coordenadas de um vetor com relac¸a˜o a uma base ordenada e matriz mudanc¸a de base. 1. v2 = (1, 1). 2. (a) E´ base; (b) Na˜o e´ base; (c) Na˜o e´ base. 3. (x, y) = (2x+ 3y)(2,−1) + (x+ 2y)(−3, 2). 4. (a) Na˜o e´ base; (b) E´ base; (c) E´ base. 5. 6. {v1, v2, v3}. 7. {v1, v2}. 8. (a) [v]B1 = [ 2 1 ] ; (b) [v]B2 = [ −23 10 3 ] ; (c) [v]B3 = [ 6 2 ] ; (d) [v]B4 = [ 2 6 ] . 9. [v]B = 2−2 −3 . 10. (a) {(1, 3, 0), (0, 0, 1)} e dim(V1) = 2; (b) {(1, 5, 0)} e dim(V2) = 1; (c) {(1,−1)} e dim(V3) = 1; (d) {(3, 1,−1)} e dim(V4) = 1; (e) {(1, 0, 0), (0, 1, 0)} e dim(V5) = 2. 11. dim(W ) = 2. 12. (a) [I]B1B2 = 0 1 01 2 −1 12 1 2 0 −12 ; (b) [v]B2 = 12 −1 ; 1 (c) [I]B2B1 = 1 1 11 0 0 1 1 −1 . 13. (a) [I]B1B2 = 1 1 10 −1 0 0 0 −1 ; (b) [v]B2 = 6−1 −2 ; (c) [I]B2B1 = 1 1 10 −1 0 0 0 −1 . 14. B = {(1, 5), (0, 6)}. 15. C = { 2 3 + 1 3x− 13x2, 13 + 23x+ 13x2 } . 2
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