Question

For each given situation, determine whether B is in the column space of A and state the consistency of the system Ax = B.

Answer 1

In this case,
`B= \left[\begin{array}{ccc}4\\8\\\end{array}\right]` is indeed in the column space of A and we can conclude that the system Ax = B is consistent.

Step-by-step explanation:

To determine if b is in the column space of A, we can check if the equation Ax = b has a solution.

Let's calculate the column space of A. The column space of a matrix A is the set of all possible linear combinations of the columns of A. In other words, it is the span of the columns of A.

Given
`A=\left[\begin{array}{ccc}1&2 \\2&4\\\end{array}\right]` and
`B= \left[\begin{array}{ccc}4\\8\\\end{array}\right]`, we can rewrite the equation Ax = B as a system of equations:

1x + 2y = 4

2x + 4y = 8

We can simplify the second equation by dividing it by 2:

x + 2y = 4

Notice that the first equation is a multiple of the second equation. This means that the two equations are equivalent and represent the same line in the xy-plane.

Since the equations are equivalent, they have infinitely many solutions and the system is consistent. Therefore,
`B= \left[\begin{array}{ccc}4\\8\\\end{array}\right]` is in the column space of A.

In summary:

-
`B= \left[\begin{array}{ccc}4\\8\\\end{array}\right]` is in the column space of A.

- The system Ax = B is consistent.

Your question is incomplete, but most probably the full question was:

For each of the following choices of A and b, determine whether b is in the column space of A and state the consistency of the system Ax = B.

`A=\left[\begin{array}{ccc}1&2 \\2&4\\\end{array}\right]`

`B= \left[\begin{array}{ccc}4\\8\\\end{array}\right]`