Practice question · Multiple choice
The same matrix equation Ax = b describes a system of equations, a transformation applied to a vector, and a combination of A's columns. Why is having three readings of one equation useful rather than merely a curiosity?
Hints
- Ask which reading makes "does a solution exist?" easiest to see.
- Now ask which one makes "why are there infinitely many?" easiest.
Show the answer
D. Because each reading makes a different question easy.
Why
The three readings are the same mathematics from different vantage points, and each makes a different question easy. Column reading answers existence at once, a solution exists precisely when b lies in the span of the columns. Transformation reading answers uniqueness, since anything in the kernel can be added to a solution. Fluency in switching is most of practical linear algebra.
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