Help to Integral Calculus Solutions & Examples

To determine:
a) To graph: The function ${\displaystyle f\left(x\right)=x^2+1}$ in the region ${\displaystyle x\le 0}$
b) Whether the function ${\displaystyle f\left(x\right)=x^2+1}$ in the region ${\displaystyle x\le 0}$ is a one-to-one function with the help of graphical representations.

Confusion about properties of linear transformation.
I learned from 3Blue1Brown's Linear Algebra videos that one of the rules for a transformation to be linear is that the origin remains fixed in place after transformation. So if ${\displaystyle T\left(x\right)=x+a}$ is a given transformation, I know that by inserting ${\displaystyle x=0}$, it is not a linear transformation. I am not able to understand what is non-linear about it.

Proof properties of Walsh Transformation
In our lecture notes we're given some properties of Walsh-Transform, for a given mapping ${\displaystyle f:F_2^n\to F_2}$ (from n-dimensional-Vectorspace over F2 to F2)
The Walsh-Transform itself is given as ${\displaystyle f^w\left(a\right)=\sum _{x\in \mathbb{F}}\left({(-1)}^{\begin{array}{cc}f\left(x\right)+<a& x>\end{array}}\right\}}$ with being the inner product of the two vectors a,x. (means ${\displaystyle \sum _i^n{a}_i\cdot x_i}$)

Problem with showing properties of linear transformation
Let ${\displaystyle F:K^3\to K^3}$ be a linear transformation, such that
1. ${\displaystyle (F-\lambda I)(F-\mu I)\ne 0}$,
2. ${\displaystyle {(F-\mu I)}^2\ne 0}$,
3. ${\displaystyle (F-\lambda I){(F-\mu I)}^2=0}$.
I want to show that
1. ${\displaystyle \lambda ,\mu }$ are eigenvalues of F,
2. ${\displaystyle ker(F-\mu I)\subseteq Im(F-\mu I)}$,
3. F has in a certain basis matrix
$(\begin{array}{ccc}\lambda & 0& 0\\ 0& \mu & 1\\ 0& 0& \mu \end{array})$

2D Signal Properties of Fourier Transformation
I was asked this question below.
I have to match i-v to a-j (matching could be one to many or many to one) and i should prove the matchings.
If conjugation is not made, i can assume that ${\displaystyle h(n_1,n_2)}$ is real.
How can i start solving this question?

Creating a transformation matrix
This question is related to this one, but more specific. I was given the following question:
Let ${\displaystyle D=\{d_1,d_2\}\text{and}B=\{b_1,b_2\}}$ be bases for vector spaces V and W respectively. Let ${\displaystyle T:V\to W}$ be a linear transformation with the property that ${\displaystyle T\left(d_1\right)=-5b_1-7b_2\text{and}T\left(d_2\right)=9b_1+4b_2}$. Find the matrix for T relative to D and B.
This is a pretty simple question and I just took the transformation properties and made a matrix:
$(\begin{array}{cc}-5& -7\\ 9& 4\end{array})$
The answer key has the same answer but with the rows and columns switched:
$(\begin{array}{cc}-5& 9\\ -7& 4\end{array})$
Are these both correct?

Need help understanding the scaling properties of the Legendre transformation
At the wikipedia page for the Legendre transformation, there is a section on scaling properties where it says
${\displaystyle f\left(x\right)=ag\left(x\right)\to f\cdot \left(p\right)=ag\cdot \left(\frac{p}{a}\right)}$
and ${\displaystyle f\left(x\right)=g\left(ax\right)\to f\cdot \left(p\right)=g\cdot \left(\frac{p}{a}\right)}$
where f*(p) and g*(p) are the Legendre transformations of f(x) and g(x), respectively, and a is a scale factor.
Also, it says:
"It follows that if a function is homogeneous of degree r then its image under the Legendre transformation is a homogeneous function of degree s, where ${\displaystyle \frac{1}{r}+\frac{1}{s}=1}$.
1) I don't see how the scaling properties hold. I'd appreciate if someone could spell this out for slow me.
2) I don't see how the relation between the degrees of homogeneity (r,s) follows from the scaling properties. Need some spelling out here too.
3) If the relation ${\displaystyle \frac{1}{r}+\frac{1}{s}=1}$ is true, then could this be used to prove that linearly homogeneous functions are not convex/concave? (Because convex/concave functions have a Legendre transformation, and ${\displaystyle r=1}$ would imply ${\displaystyle s=\mathrm{\infty }}$, which is absqrt and thus tantamount to saying that a function with ${\displaystyle r=1}$ has no Legendre transformation. No Legendre transformation then implies no convexity/concavity.)

Mobius transformation satisfying certain properties
I'm having some trouble showing that a Mobius transformation F maps 0 to ${\displaystyle \mathrm{\infty }\text{and}\mathrm{\infty }}$ to 0 iff ${\displaystyle F\left(z\right)={dz}^{-1}}$ for some ${\displaystyle d\in \mathbb{C}}$. Mainly with the "only if" part. Do I need to use pictures?

If a linear transformation has any two of the properties of being self-adjoint, isometric, or involutory, [closed]
Prove that if a linear transformation has any two of the properties of being self-adjoint, isometric, or involutory, then it has the third. (Recall that an involution is a linear transformation A such that ${\displaystyle A^2=1}$.)

Let T be a linear transformation from ${\displaystyle R^2\text{into}{R}^2}$ such that ${\displaystyle T(1,0)=(1,1)\text{and}T(0,1)=(-1,1)}$. Find ${\displaystyle T(6,1)\text{and}T(-5,1)}$
${\displaystyle T(6,1)=?}$
${\displaystyle T(-5,1)=?}$

Linear transformation and its

Do either of the linear transformation properties imply the other?
Im

Properties of ${\displaystyle A^{T}A}$ transformation.
Is it true a proposition for ${\displaystyle 3\times 3}$ real matrices A that if for a some nonzero vector v we have ${\displaystyle Av\ne v}$ and at the same time ${\displaystyle \left(A^{T}A\right)v=v}$ then for any vector u from ${\displaystyle {\mathbb{R}}^3}$ we have ${\displaystyle \left(A^{T}A\right)u=u}$ (i.e. A is orthogonal) ?
Do we need assumption about full rank of the matrix A for this goal?
Can we extend the proposition for bigger dimensions?

Constructing a transformation satisfying the given properties
Construct a transformation which maps the plane into the plane itself , and the curve ${\displaystyle y=x^2}$ onto the horizontal axis, and the line ${\displaystyle y=3}$ onto the vertical axis.
I need to solve the set of equations
${\displaystyle T(x,x^2)=(x,0)}$
${\displaystyle T(x,3)=(0,y)}$
but couldn't. Any hints?

Rational scalar multiplication in Linear transformation using addition property [duplicate]
Linear transformation: Map ${\displaystyle T:V\to W}$ is said to be Linear transformation when it satisfy property:
1) ${\displaystyle L(u+v)=L\left(u\right)+L\left(v\right)\mathrm{\forall }u,v\in V}$
2) ${\displaystyle L\left(cv\right)=cL\left(v\right)\mathrm{\forall }c\in \mathbb{F}}$
I know when ${\displaystyle c\in \mathbb{N}}$ then property 2 can be obtained from property 1, since
${\displaystyle L\left(nv\right)=L\left(v\right)+L\left(v\right)\dots ..+L\left(v\right)}$ (n times where ${\displaystyle n\in \mathbb{N})}$
Now I have read in somewhere that if our scalar c is rational then property 2 also follows from property 1, but I am unable to prove this. How can I show that if ${\displaystyle c\in \mathbb{Q}}$, when (2) follows from (1)?
I know that if our scalar is irrational, then we cannot use property 1 to prove property 2.

Proof of a linear transformation property
Suppose ${\displaystyle \varphi :X\to Y}$ is a map of sets and F is a field. Let ${\displaystyle \varphi \cdot :F\left(Y\right)\to F\left(X\right)}$ be a map sending a function ${\displaystyle f\in F\left(Y\right)}$ to a function ${\displaystyle \varphi \cdot \left(f\right)\in F\left(X\right)}$ given by ${\displaystyle \varphi \cdot \left(f\right)\left(x\right)=f\left(\varphi \left(x\right)\right)}$ for every ${\displaystyle x\in X}$.
How can we prove that for a scalar ${\displaystyle \lambda \in F,\lambda \cdot \varphi \cdot \left(f\right)=\varphi \cdot \left(\lambda f\right)}$?
Take it as axiomatic that ${\displaystyle \varphi \cdot }$ is a ring homomorphism if needed.

Determine the equation of the function with the following properties:
A transformation of the basic ${\displaystyle y=\cot \left(x\right)}$ function with
- A period of ${\displaystyle \frac{\pi }{3}}$
- A stretching factor of 3
- A phase shift to the left by straight ${\displaystyle \frac{\pi }{12}}$
- A vertical shift down by 5
Write the equation for the function described above:

If P is linear, then
1) None of the given options is correct
2) The null space of P is empty
3) ${\displaystyle P(x-y)=P(x+y)}$ for all inputs x,y
4) ${\displaystyle P(ax+by)=aP\left(x\right)+bP\left(y\right)}$ for all inputs x, y and constants a, b.

Is the transformation ${\displaystyle R_{\frac{\pi }{3}}\left(x\right)}$ is linear transformation? Justify your answer.

Properties of dual linear transformation
Let V,W finite dimensional vector spaces over the field F, and let ${\displaystyle T:V\to W}$ a linear transformation.
Define ${\displaystyle T\cdot :W\cdot \to V\cdot }$ the dual linear transformation. i.e ${\displaystyle T\cdot \left(\psi \right)=\psi \circ T}$.