Why are these 2 algebraic expressions equivalent February 14    2

I just solved a long problem for my physics w/calculus homework that required a simplification using a quadratic formula. The "textbook" (flipItPhysics) came up with a different simplification than mine but it turns out they are equivalent. I ca

Quadratic expression February 14    1

If $$\frac{a_0}{n+1}+\frac{a_1}{n}+\frac{a_2}{n-1}+\ldots+\frac{a_{n-1}}{2}+a_{n}=0,$$ then the maximum possible number of roots of the equation $${a_0}{x^n}+{a_1}{x^{n-1}}+{a_2}{x^{n-2}}+\ldots+{a_{n-1}}{x}+{a_n}=0$$ in $(0,1)$ will be...?This seems

Finding orthonormal basis February 14    1

I need to find an orthogonal basis for certain symmetric bilinear form $g$ such that matrix of g is diagonal in these basis.Given basis ${b_1,b_2,b_3}$I am applying the following process (called Gramm Schmidt I guess, but I am confused because doesnt

Finding the orthogonal basis, picture included February 14

I decided to share a picture of what I have so far. I am not sure if I did it correctly and sorry if it is not readable. Ask me if anything is unclear. In the exercise I am basically just asked to find the ortogonal basis. I did it by finding the red

How to calculate the curvature of a curve whose equation is not given. February 14    2

How to calculate the curvature of a curve whose equation is not given.
I want to calculate the curvature of the following curve (in blue) whose equation is not known. I shall be thankful.This looks like an ellipse: $\left(\frac x4\right)^2+\left(\frac y3\right) ...

Metric Spaces Whose Diameter is Achieved at Every Point. February 14

Metric Spaces Whose Diameter is Achieved at Every Point.
Suppose $(X,d)$ is a metric space with diameter $\sup \{ d(x,y) \colon x,y \in X\}=1$. Call the point $x \in X$ an edge point to mean that $d(x,y)=1$ for some $y \in X$. Call the metric spac ...

Berry's curvature equation February 14

The following textbook Heisenberg's Quantum Mechanics shows an example of calculating Berry's curvature (top page on pg 518). It led to a following equation [1] $$ V_{m} = {- 1 \over B^2 } * i * \sum { (<m,B|S|n,B> ∧ <n,B|S|m,B>) \over A^2} $$

Cancellation law for invertible matricies February 14    2

Show that the cancellation law holds for invertible matrices. i.e. if $A \in GL_n(R), B, C \in M_{n×m}(\mathbb{R})$ and $AB = AC$, then $B = C$.What I tried:I know that I can prove this by actually looking at each element of the matrix and multiplyin

Orientation on a manifold as a sheaf February 14    1

Orientation on a manifold as a sheaf
I am thinking about orientation of a connected manifold $M$ of dim $n$ as a sheaf.There are two definitions I could use, the first is the sheaf associated to the presheaf $$U\mapsto H_n(M,M- ...

Let U and V be vector spaces of dimensions n and m over K. Find the dimension and describe a basis of Homk(U,V) February 14

Let U and V be vector spaces of dimensions n and m over K. Find the dimension and describe a basis of Homk(U,V)
I am given vectors spaces U and V of dimensions n and m over K.How can I find the dimension and basis of Homk(U,V) ?

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