Uni Maths Questions

[kamil9876]

I agree with both of your answers (not sure what a "symmetric closure" is but the name suggests that it is the smallest relation contain your relation that is symmetric, if that's the case then YES).

[Deleted User]

How do I do this question?    

Find a basis for the solution space of the following system in five variables:
-x1 - x2 -3*x4 - x5 = 0
3*x1 + 3*x2 - x3 -3*x4 + x5 = 0

The answer gives me a 3x5 matrix but I don't know how this is possible if there are only 2 rows in the system?

[Jeggz]

I'm having a bit of trouble with this question -
Find the distance from the point (1,0,0) to the line through the points (1,2,0) and (-1,1,1)
Any help would really be appreciated! 

[Will T]

Find the equation of the line in Parametric form:


Any point on the line has co-ordinates
From the distance formula, the distance between the line and a point will be:








Therefore, the minimum occurs when and is
Or something like that....

[Alwin]

How do I do this question?    

Find a basis for the solution space of the following system in five variables:
-x1 - x2 -3*x4 - x5 = 0
3*x1 + 3*x2 - x3 -3*x4 + x5 = 0

The answer gives me a 3x5 matrix but I don't know how this is possible if there are only 2 rows in the system?

Think of the kernal space (ie solve the system of equations)
You have 2 equations and 5 equations. Hence, 3 parameters required which gives you your 3 rows. Your 5 columns are a result of the 5 variables

I'm having a bit of trouble with this question -
Find the distance from the point (1,0,0) to the line through the points (1,2,0) and (-1,1,1)
Any help would really be appreciated! 

@Will T, your method is perfectly acceptable. However, a more elegant solution that requires less algebraic manipulation is as follows:











I find this method better because Will T's method requires a lot more work.

If you are interested in the proof of this formula, it is a generalisation of projections of vectors, ie the vector method of finding distance rather than the algebraic method.

Hope it helps!

[Phy124]

I'll get back to you on this later (if someone else doesn't beforehand) but in the mean time you've made an error in this line:

[Deleted User]

How do I show whether this is a linear transformation (or not)?
F: R^3 -> R^2, F(x,y,z) = (0,2x+y)

I know I have to prove F(u+v) = F(u) + F(v) and F(au)= aF(u) but idk how to go about doing this

[mark_alec]

F(a+x, b+y, c+z) = (0, 2a+2x+b+y) = (0, 2a+b) + (0, 2x+y) = F(a,b,c) + F(x,y,z)
F(cx, cy, cz) = (0, 2cx+cy) = c(0, 2x+y) = cF(x,y,z)

[Deleted User]

Thanks man!

[Deleted User]

How do a find a single matrix that gives this linear transformation: reflects about y-axis, then expands by factor 5 in x-direction, then reflects about y=x? Thanks

[kamil9876]

So doing B then A is equivalent to doing AB (because B takes v to Bv, then A takes Bv to A(Bv)=(AB)v)

So if you know the matrix of each of those transformations you can just multiply them together.

Another Way:

The image of (1,0) is the first column. The image of (0,1) is the second column. So let us see how (1,0) travels:

(1,0) -> (1,0) -> (5,0) -> (0,5)

(0,1) -> (0,-1) -> (0,-1) -> (-1,0)

So the matrix is:

0 -1
5  0

[Jeggz]

Find the point of intersection of the line 2x=y-1=z with the plane x+2y-z=2. What is the angle between the line and the plane?
I'm stuck guys, any help would really be appreciated 

[brightsky]

line has equation r = (0,1,0) + t*(1/2,1,1), where t E R
plane has equation r.(1,2,-1) = 2
sub the line into the plane:
(0,1,0).(1,2,-1) + t(1/2,1,1).(1,2,-1) = 2
2 + t(1/2+2-1) = 2
2 + t(3/2) = 2
t = 0
so the point of intersection is (0,1,0).

the line has direction vector d = (1/2,1,1). the plane has normal vector n = (1,2,-1).
so the cosine of the acute angle between d and n is given by:
absolute value(d.n/mod(d)mod(n)) = (1/2 + 2 - 1)/(3/2)(sqrt(6)) = sqrt(6)/6
so the acute angle is arccos (sqrt(6)/6).
now we require the angle between the line and the plane, which is, then: pi/2 - arccos(sqrt(6)/6)

[Deleted User]

Hi guys, what do these bases mean: B= {e1,e2,e3} and B'={E^(ij)|i=1,2;j=1,2}? Something to do with 2x2 matrices?

[Alwin]

I have a question and its really bugging me

prove that if all the rows of a matrix add up to the same number k, then k is an eigenvalue of this matrix. Describe one possible eigenvector corresponding to this special eigenvalue.

Okay, I got the second part, the eigenvector is a constant vector eg but any ideas about proving k is an eigenvalue??

Thanks guys!