Let A be a 5 by 4 matrix. Explain why the equation ax=b cannot be co.pdf

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Let A be a 5 by 4 matrix. Explain why the equation ax=b cannot be consistent for all be in R^5. Solution HINT: The equation Ax=b is consistent if and only if b is a linear combination of the columns of A, i.e., if and only if b is in the column space of A. What is the maximum possible dimension of the column space of A? What is the dimension of R3? (If you get the point of this hint, the generalization should be pretty obvious.).

Let A be a 5 by 4 matrix. Explain why the equation ax=b cannot be consistent for all be in R^5.
Solution
HINT: The equation Ax=b is consistent if and only if b is a linear combination of the columns of
A, i.e., if and only if b is in the column space of A. What is the maximum possible dimension of
the column space of A? What is the dimension of R3? (If you get the point of this hint, the
generalization should be pretty obvious.)

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