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LAHW#02

LAHW#02. • Due October 17, 2011. Section 1.3 Kernels, Rank, Homogeneous Equations. 4. For the matrix shown here, compute its rank and find the set of vectors whose span is its kernel:. Section 1.3 Kernels, Rank, Homogeneous Equations. 6.

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LAHW#02

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  1. LAHW#02 •Due October 17, 2011

  2. Section 1.3Kernels, Rank, Homogeneous Equations • 4. • For the matrix shown here, compute its rank and find the set of vectors whose span is its kernel:

  3. Section 1.3Kernels, Rank, Homogeneous Equations • 6. • Find a matrix whose kernel is spanned by the two vectors u = (1, 3, 2) and v = (-2, 0, 4).

  4. Section 1.3Kernels, Rank, Homogeneous Equations • 13. • Establish that if the set {v1, v2, v3} is linearly independent, then so is {v1+ v2, v2+ v3, v3+ v1}.

  5. Section 1.3Kernels, Rank, Homogeneous Equations • 16. • If and the kernel of A contains the vector , what are x, y, and z?

  6. Section 1.3Kernels, Rank, Homogeneous Equations • 35. • Let A be an m × n matrix, where m < n.What is the maximum number of pivot positions in A? What is the least number of nonpivotal column in A? What is the least number of free variables in solving the equation Ax = 0?

  7. Section 1.3Kernels, Rank, Homogeneous Equations • 36.(Continuation.) • Adopt the hypotheses on A as in the preceding question. Explain why the equation Ax = 0 has a nontrivial solution. Explain why the columns of A form a linearly dependent set of vectors.

  8. Section 1.3Kernels, Rank, Homogeneous Equations • 41. • If the rank of an augmented matrix [A|b] is greater than the rank of A, what conclusion can be drawn? Is there an implication in both directions?

  9. Section 1.3Kernels, Rank, Homogeneous Equations • 61. • Explain that if the equation AX = B has more than one solution, then the equation AX = 0 has a nontrivial solution. Establish that this equation has infinitely many solutions. Here A is an m × n matrix, X is an n × q matrix, and B is an m × q matrix.

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