# Linear Algebra - Matrix Notation

Consider the matrix $$A = \begin{pmatrix}3 & 5 & 11 \\5 & 9 & 20\\11&20&49\end{pmatrix}$$

I have attached an image of the question here.

Could someone please solve the first question, that is the magnitude of $v_2$ . I am not sure which column the vectors belong to, thus I am unable to solve any of the parts. $$\begin{pmatrix}v_1v_1 &v_1v_2 & v_1v_3\\ v_2v_1 &v_2v_2& v_2v_3\\ v_3v_1 &v_3v_2 &v_3v_3\end{pmatrix}$$

• Here you will find a MathJax tutorial for future questions. Also I think you should read this. Jul 10, 2016 at 18:16
• @AaronMaroja Thanks. I also figured out how to solve the problems. Jul 10, 2016 at 18:19

The problem says that the entries of the matrix are given by the dot product of the vectors in $\mathbb R^4$.

For instance, $$3 = a_{11} = \vec v_1 \cdot \vec v_1 \\5 = a_{12} = \vec v_1 \cdot \vec v_2\\11 = a_{13} = \vec v_1 \cdot \vec v_3$$ and so on.

Here, the matrix $A$ has the form $$A = \begin{pmatrix}a_{11} & a_{12} & a_{13}\\a_{21} & a_{22} & a_{23}\\a_{31} & a_{32} & a_{33}\end{pmatrix}$$

Extra: Here are some useful hints:

i) $\|\vec v_2\|^2 = \vec v_2 \cdot \vec v_2$;

ii) $\|\vec v_1 + \vec v_2\| \leq \|\vec v_1\| + \|\vec v_2\|$;

iii) $\cos \theta = \frac{\langle \vec v_2, \vec v_3\rangle}{\|\vec v _2\| \|\vec v_3\|}$, where $\theta$ is the angle between $\vec v_2$ and $\vec v_3$.

Just a few ideas so you can get started.

• I understand that, but if we for instance asked to compute the magnitude of v2 i would think that we would have to find the length of the first column, but this is not correct. So i am not sure how the dot product between the two vectors ties to just v2 for instance. Jul 10, 2016 at 17:52
• So the problem isn't about notation. You want to solve your question. You should rephrase it then. Jul 10, 2016 at 17:53
• I believe the issue is that I forgot that the dot product of a vector with itself gives its length. Jul 10, 2016 at 18:05
• @Irina That's not true. It gives you something related to its length, but not the length itself. Jul 10, 2016 at 18:06
• @MatthewDrury my mistake, it is the square of the length, thus in order to find the length, do we square root it, here it would be 3. Jul 10, 2016 at 18:11
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