Q1. If vector v and w are orthogonal unit vectors, how can I calculate the length of (v-2w) ??
Q2. Is it correct that for a matrix A and w=2u-3v, Aw=2Au-3Av??
Q3. Is the following sentence true? : If columns 1 and 2 of B are the same, then so are columns 1 and 2 of AB.
If not, what would be the correct sentence??
The length of any vector $u$ is the square root of the dot product with itself, or $\sqrt{u \cdot u}$.
For Q1, replace $u$ above with $v-2w$, and use the distributive properties of dot product to expand it. Since $v$ and $w$ are orthogonal, $v \cdot w=0$. Since $v$ and $w$ are both unit vectors, they each have magnitude 1.
Q2) Yes.
Q3) Think about how you would get columns 1 and 2 of $AB$. To get column 1, for example, you would do the dot product of the first row of $A$ with the first column of $B$.