# Help with basic linear algebra

1. Jan 24, 2008

### andrassy

1. The problem statement, all variables and given/known data Theres two questions I need help with on m homework:

I need to prove algebraically that the linear system r + 2s = -b1 and 3r+5s = b2 has a solution for all numbers b1, b2

also: for vectors v and w prove that v-w and v+w are perpendicular if and only if the magnitude of v equals the magnitude of w.

3. The attempt at a solution The first question i multiplied the first equation by -3 and added the two equations together to get 11s = b2 - 3b1 but i have no idea where to go from there.

The second I proved that two vectors are perpendicular if their dot product is zero. I did the dot product of v-w and v+w an dgot [v1^2 - w1^2, . . . vn^2 - wn^2].here agian in stuck. any help please?

2. Jan 24, 2008

### Defennder

When you say prove algebraically, are you allowed to use matrices here? It's a lot easier if you could do so. Start by representing the linear system as an augmented matrix:

$$\left(\begin{array}{*{20}c}1&2&-b_{1}\\3&5&b_{2}\end{array}\right)$$

If you want to show that there are exactly 1 solutions for both r,s , you need to show that you can reduce the augmented matrix (the sub-matrix on the left) above to the identity matrix.

For the 2nd part, your approach is correct, but you should get this:

$$(v+w)\cdot(v-w) = v\cdot v + v\cdot w - w\cdot v - w\cdot w$$

You know where to go from here, right?

3. Jan 24, 2008

### Rainbow Child

You are almost done. Solve for s, and sustitude the result in one of original the equations in order to find r.

Write the dot product $(\vec{v}-\vec{w})\cdot (\vec{v}+\vec{w})=0$ and expand it.