Use the theory in section 4.6 to help you solve these problems using words without calculations....

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Use the theory in section 4.6 to help you solve these problems using words without calculations. Citing theorem 4.19 on page 204 will help with some of these questions. Also, keep in mind the nullspace(A), columnspace(A), and rowspace(A) are all vector spaces (or subspaces of the appropriate R"). 1 Solutions of Linear Systems Write a short paragraph to answer each of the following questions about solutions of systems of linear equations. You should not perform any calculations, but instead base your explanations on the appropriate properties from the text. 1. One solution of the homogeneous linear system x + 2y + 2+ 3w=0 x- y + w=0 y-z+2w=0 is x = -2, y = -1, z = 1, and w = 1. Explain why x = 4, y = 2, z = -2, and w = -2 must also be a solution. Do not perform any row operations. 2. The vectors x, and x₂ are solutions of the homogeneous linear system Ax = 0. Explain why the vector 2x₁ - 3x2 must also be a solution. 3. Consider the two systems represented by the augmented matrices. 1 -5 0-2 31 JE 1-5-91 0-2-3 1 2 -1 -1 If the first system is known to be consistent, explain why the second system is also consistent. Do not perform any row operations. 0 4. The vectors x₁ and x₂ are solutions of the linear system Ax = b. Is the vector 2x₁ - 3x₂ also a solution? Why or why not? 5. The linear systems Ax=b₁ and Ax=b₂, are consistent. Is the system Ax = b₁ + b₂ necessarily consistent? Why or why not? 6. Consider the linear system Ax= b. If the rank of A equals the rank of the augmented matrix for the system, explain why the system must be consistent. Contrast this to the case in which the rank of A is less than the rank of the augmented matrix. Use the theory in section 4.6 to help you solve these problems using words without calculations. Citing theorem 4.19 on page 204 will help with some of these questions. Also, keep in mind the nullspace(A), columnspace(A), and rowspace(A) are all vector spaces (or subspaces of the appropriate R"). 1 Solutions of Linear Systems Write a short paragraph to answer each of the following questions about solutions of systems of linear equations. You should not perform any calculations, but instead base your explanations on the appropriate properties from the text. 1. One solution of the homogeneous linear system x + 2y + 2+ 3w=0 x- y + w=0 y-z+2w=0 is x = -2, y = -1, z = 1, and w = 1. Explain why x = 4, y = 2, z = -2, and w = -2 must also be a solution. Do not perform any row operations. 2. The vectors x, and x₂ are solutions of the homogeneous linear system Ax = 0. Explain why the vector 2x₁ - 3x2 must also be a solution. 3. Consider the two systems represented by the augmented matrices. 1 -5 0-2 31 JE 1-5-91 0-2-3 1 2 -1 -1 If the first system is known to be consistent, explain why the second system is also consistent. Do not perform any row operations. 0 4. The vectors x₁ and x₂ are solutions of the linear system Ax = b. Is the vector 2x₁ - 3x₂ also a solution? Why or why not? 5. The linear systems Ax=b₁ and Ax=b₂, are consistent. Is the system Ax = b₁ + b₂ necessarily consistent? Why or why not? 6. Consider the linear system Ax= b. If the rank of A equals the rank of the augmented matrix for the system, explain why the system must be consistent. Contrast this to the case in which the rank of A is less than the rank of the augmented matrix.

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