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Right, is this better? Also, do I need to give reasons at each step like you do?

Thanks! Yeah, just realised that I violated order of operations lol. x I'll get back to figuring out the second part of (1). Would this be right for (3) though? EDIT: the first step is meant to have brackets around the whole thing, i.e.

Is this right? This is for the first question, btw

Thanks But is there a way to prove the distributive laws WITHOUT considering the matrices component wise?

Thanks, I'll leave the latter for later! I'm pretty frustrated by these...What would be the shortest way of doing them? Prove the following properties of matrices: 1) If the product AB exists, then A(λB) = λ(AB) = (λA)B 2) Associative law of matrix multiplication. If products AB and BC exist...

Thanks leehuan! And yeah you're right, my notation is wrong for the augmented matrix, appreciate that. I have another question: $\noindent 1. Prove that $|Ax-y|$ is least when $y-Ax$ is orthogonal to $\sigma =$span(columns of $A).$ \text{Given a set of vectors}...

Thanks, it is indeed easier! I found my mistake, so when I calculate DE = (1,1,1) and EF = (1,1,2), I find cos(DEF) = 4/sqrt(18). Previously, I found 4/(sqrt(12). I feel so dumb.... Thanks! Also, there isn't an answer provided at the back of the problems book for this, could you please...

Thanks. $\noindent Let D, E, F be the points with coordinate vectors$\\d = \begin{pmatrix}5\\6\\7\end{pmatrix}, e = \begin{pmatrix}6\\7\\8\end{pmatrix}, f = \begin{pmatrix}7\\8\\10\end{pmatrix} \\$(a) Caculate $cos(\angle DEF)$ as a surd.$\\$(b) Calculate the area of $\Delta DEF$ as a surd.$...

Thanks $\noindent Convert $r=6sin \theta (0\leq \theta \leq \pi)$ into Cartesian form. Do this for $ r=2cos \theta (-\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2}).

Aaah...found them, thanks again! $\noindent 16. Let $B$ be a point in $\mathbb{R}^{n}$ with coordinate vector b. Let $x = a + \lambda d, \lambda \in \mathbb{R}$ be the equation of a line. $\\$(a) Show that the square of the distance from B to an arbitrary point x on the line is given by...

As far as I remember, the OSI model is covered in the syllabus notes on the official BOSTES website (https://www.boardofstudies.nsw.edu.au/syllabus_hsc/pdf_doc/ipt-st6-syl-from2010.pdf - refer page 36. Also, Davis has a pretty in-depth section about OSI model (can't recall the page number tho...)

Thanks InteGrand. I've another question: $\noindent 1. Urn 1 contains 2 red balls and 3 black balls. Urn 2 contains 4 red balls and 5 black balls.$\\$a) Suppose a ball is drawn at random from urn 1 and placed into Urn 2. If a ball is then drawn at random from the 10 balls in urn 2, what is the...

I had another question: $\noindent A brewery brews one type of beer which is marketed under three different brands. In a survey of 150 first year students, 58 drink at least Brand A, 49 drink at least Brand B and 57 drink at least Brand C. 14 drink brand A and brand C, 13 drink brand A and...

I agree. And @leehuan, yes InteGrand is a great guy!

Wow, thanks!

Thanks Sorry just fixed that up, it was a typo lol!

\\\text{a) }E(X) = \sum_{k=1}^{\infty} kp_{k}\\\therefore E(X^{2}) = \sum_{k=1}^{\infty} k^{2}p_{k}\\=\sum_{k=1}^{\infty} \frac{k^{2}e^{-\lambda}\lambda^{k}}{k!}\\=\sum_{k=1}^{\infty} \frac{ke^{-\lambda}\lambda ^{k}}{(k-1)!}\\\text{b) Let }l = k-1\\\therefore E(X^{2})= \sum_{l=0}^{\infty}...

Wow, thanks! I had another question: \\\text{Let X be a discrete random variable with Poisson distribution with parameter }\lambda\\\text{a) Show that }\\ E(X^{2})=\sum_{k=1}^{\infty}\frac{ke^{-\lambda}\lambda^{k}}{(k-1)!}\\\text{b) By changing the summation variable, show that...

Could you please demonstrate how one would go about doing this?

Thanks for the reply InteGrand. I was thinking, isn't there a method to do this using dot products and/or cross product? The answer to this question was the point (3,2,2), which just happens to be the point in the line equation...