Algebra and Geometry | Part IA, 2005

Let A\mathbf{A} be a real 3×33 \times 3 matrix. Define the rank of A\mathbf{A}. Describe the space of solutions of the equation

Ax=b,(†)\tag{†} \mathbf{A x}=\mathbf{b},

organizing your discussion with reference to the rank of A\mathbf{A}.

Write down the equation of the tangent plane at (0,1,1)(0,1,1) on the sphere x12+x22+x32=2x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=2 and the equation of a general line in R3\mathbb{R}^{3} passing through the origin (0,0,0)(0,0,0).

Express the problem of finding points on the intersection of the tangent plane and the line in the form ()(†). Find, and give geometrical interpretations of, the solutions.

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