Question

Please help: linear algebra problem. (Linear combinations)

Answer 1

`\left[\begin{array}{ccc}5&7\\5&-8\\3&-9\end{array}\right] \left[\begin{array}{ccc}c\\d\end{array}\right] =\left[\begin{array}{ccc}-16\\3\\-15\end{array}\right]`

This tells us that:

`A=\left[\begin{array}{ccc}5&7\\5&-8\\3&-9\end{array}\right]`

`b=\left[\begin{array}{ccc}-16\\3\\-15\end{array}\right]`

Explanation:

So we are saying we have scalars, c and d, such that:

`c\left[\begin{array}{ccc}5\\5\\ 3\end{array}\right]+d\left[\begin{array}{ccc}7\\-8\\-9\end{array}\right]=\left[\begin{array}{ccc}-16\\3\\-15\end{array}\right]`.

So we want to find a way to express this as:

Ax=b where x is the scalar vector,
`\left[\begin{array}{ccc}c\\d\end{array}\right]`.

So we can write this as:

`\left[\begin{array}{ccc}5&7\\5&-8\\3&-9\end{array}\right] \left[\begin{array}{ccc}c\\d\end{array}\right] =\left[\begin{array}{ccc}-16\\3\\-15\end{array}\right]`