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Apr
13
comment Generating data for overdetermined least square problem
By consistent, I mean that Ax = b holds even though it is over-determined. Your approach generates a $b$ which is consistent with $A$. This is not typically the case for least squares problems.
Apr
13
comment Generating data for overdetermined least square problem
Do you need A and b to be consistent? What have you tried?
Feb
26
comment Are orthogonal spaces exhaustive, i.e. is every vector in either the column space or its orthogonal complement?
@Ethan, Yes, you are correct that the union (in the formal sense) between the two subspaces is zero. I took the sense of your question at the end of the first paragraph "are there some vectors in $R^n$ which are in neither?" I thought that by using the term union, you intended span.
Feb
26
comment Are orthogonal spaces exhaustive, i.e. is every vector in either the column space or its orthogonal complement?
Yes, see this: en.wikipedia.org/wiki/Fundamental_theorem_of_linear_algebra
Feb
25
comment How To Generate Random Points on the Positive Side of a Plane in 3-D
See my second edit.
Feb
25
comment How To Generate Random Points on the Positive Side of a Plane in 3-D
This link tells you how to do it.
Feb
21
comment Integral of $e^{\overline{z}}$
Try using separate integrals over the top and bottom halves of the unit circle. I.e. one integral from $0$ to $\pi$ and another from $\pi$ to $2\pi$.
Feb
21
comment Area of a triangle with length of one side
Is angle ABC the right angle or one of the other two?
Feb
20
comment Double integral of a region.
You have the right thought. See my second edit coming shortly....
Feb
20
comment Double integral of a region.
I don't think that your solution for shape 2 is correct. Note that although the shape is symmetrical, there is no indication that the function is.
Feb
13
comment Unreachable rubik cube positions.
The OP explicitly stated "by applying standard moves", so moving stickers doesn't count. This answer does not address the intent of the OP.
Feb
12
comment Linear Algebra - Show that $M$ is not a vector space
So $\begin{bmatrix}0&b\\c&d\end{bmatrix}$ and $\begin{bmatrix}a&b\\c&0\end{bmatrix}$ are in the set. Is their sum in the set?
Feb
12
comment Linear Algebra - Show that $M$ is not a vector space
Is the set closed under addition?
Feb
9
comment a and b are integers where gcd(a,b)=p which is a prime. find gcd(a^2,b^2).
By squaring a and b, you do not introduce any additional prime factors.
Feb
8
comment What is does the transformation $[\mathbf{a}]_{\times}$ do?
Aside from its association with the cross product, it is generally known as a Skew-Symmetric Matrix which has several interesting properties. Generally rotations can be encoded in exponental form using these matrices, generally $\mathbf{R}(\mathbf a, \theta) = \exp(\theta[{\mathbf {a}}]_{{\times }})$ where $\theta$ is the angle of rotation, and $\mathbf a$ is a unit vector representing the axis.
Feb
7
comment Physics Vector Problem - Airplane
@JohnHabert, Yes, of course. Good catch... should have sketched it out.
Feb
7
comment Physics Vector Problem - Airplane
It is like the Pythagorean theorem but for general triangles. It is ideal to use when you know two sides and the angle between them to find the third side. In your case the two sides are your two lengths, the angle is 70 degrees.
Feb
7
comment Physics Vector Problem - Airplane
Do you know the law of cosines?
Feb
7
comment Basic Euclidean Geometry, Circle Arc
Your strategy would depend on the relative distance between the way points and the turning radius of your vehicle. If the distance is much larger than the radius, then you should just point and shoot for the next way point (i.e. drive a straight line), if they are of the same order of magnitude you will probably need to compute curve paths.
Feb
2
comment Determining whether the system will have a nontrivial solution?
Perhaps you could explain what a1, a2, and a3 are. If they represent columns or rows of the matrix, then it will be singular because they are linearly dependent.