3 Suppose matrix A is 5x4 Explain why the equation Ax b can

3. Suppose matrix A is 5x4. Explain why the equation Ax b can not be consistent for every b in R 4. Suppose matrix A is 4x5 and has 4 pivot positions. Do the columns of A span R Explain.

Solution

3)

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 R5?

The maximum possible dimension of A is cartesian space (4). The dimension of R5 is 5D space (5).
Therefore, since the dimensions are not equal, there is no way that Ax=b could be consistent for all b

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4)

First of all, lets check if the system is consistent.
The system must be consistent. Because the matrix
is the COEFFICIENT matrix and each row has a pivot, it means
for the AUGMENTED matrix no row is of the form (0, · · · , 0, b).
CAUTION: you can not argue that there are free variables, so the system is consistent.

Now, whenever the system is consistent, it must be in the span

So, yes, columns of A will span R4