In Exercises 1-8, find a basis for (a) the column space and - O (b) the null space of each matrix. W N 5. to w is W N - - - 1. 2 - W UN NOW- 4. NEWN 3 . 8. NOW NN 2 -3 4.2 Basis and Dimension 251 In Exercises 9-16, a linear transformation T is given. (a) Find a basis for the range of T. (b) If the null space of T is nonzero, 22 ER: -x1 + .x2 + 2x3 =0 and find a basis for the null space of T. *1+ 2x2+ 13 9. = 2x1 + 3x2 + 3.x3 2x1 - 3x2 + 4x3 = 0 *1+ 2x2 + 413_ [x1 + 212 - 13 10. 7 = XI + X2 23 X2 12 - X3 X3 E R4: x - 2x2 + 3x3 - 4x4 = 0 XA *1 - 2x2 + *3 + $4 11. 7 - 2x1 - 5x2 + x3 + 3x4 24 *2 X1 - 3x2 + 214 E R*: x1 -x2 + 213 +.14 = 0 and [x1 + 213 + X4 12. 7 = X + 3x3 + 2x4 2x1 - 3x2 - 513 - 14 =0 -x1 + 13 *1 +12 + 2x3 - X4 25. Span 13. 7 2x1 + 12 + 13 = 3x1 + 12 +.14 26. Span -2x1 - 12 + X4 14. 7 *1 + 212 + 3x3 + 4x4 27. Span 2x1 + 3x2 + 4x3 + 5x4 *1+ 212 + 3.x3 + 4.x5 15. T = 3x1 + 2 - X3 - 3x5 28. span - . ] 7x1 + 4x2 + 13 - 2x5 29. Span -XI + x2 + 4x3 + 6x4 +9x5 16. T XI + *2+ 2x3 + 4x4 + 3x5 3.x1 + *2 + 2x4 - 3.x's X1 + 2x2 + 5x3 + 9x4 + 9x5_ 30. Span In Exercises 17-32, find a basis for each subspace. 17. _25 ER : s is a scalar 2s 18, -$ + 41 E R3 : s and : are scalars 8 - 31 32. Span 5r - 3s 19 2r 0 E R4 : r and s are scalars -4s T&F In Exercises 33-52, determine whether the state- Sr - 3s ments are true or false. 2r + 65 33. Every nonzero subspace of R" has a unique basis. 20 4s - 71 E R*: r, s, and r are scalars 34. Every nonzero subspace of R" has a basis. 3r - $+91 35. A basis for a subspace is a generating set that is as large as possible. 21 ER': X1 - 3.12 + 5.x3 = 0 36. If S is a linearly independent set and Span S = V, then S is a basis for V