Explain how to find the value of sin 390^{\circ} using periodic properties.

Guided Proof Let $v_1,v_2,....V_n$ be a basis for a vector space V.
Prove that if a linear transformation $T:V\to V$ satisfies
$T(v_i)=0fori=1,2,...,n,$ then T is the zero transformation.
To prove that T is the zero transformation, you need to show that $T(v)=0$ for every vector v in V.
(i) Let v be the arbitrary vector in V such that $v=c_1{v}_1+c_2{v}_2+\cdots +c_n{V}_n$
(ii) Use the definition and properties of linear transformations to rewrite T(v) as a linear combination of $T(v_j)$ .
(iii) Use the fact that $T(v_j)=0$
to conclude that $T(v)=0,$ making T the zero transformation.

Analyze ${\displaystyle fx\left(x\right)=4\sin (3x-6)-2}$ using our “trig language” (amplitude, period, etc.) and relate each trig feature to the transformational analysis we’ve done throughout the semester. In other words, state both a transformation, such as shifted left 118 units, and the corresponding “trig language,” phase shift of 118 units left. Be sure to include all transformations and trig features.

A function f (t) that has the given Laplace transform F (s).

Find and describe an example of a matrix with each of the following properties.
Briefly describe why the desired property is in the matrix you picked. If no such matrix exists, explain using a theorem studied.
(a) A square matrix representing an injective but not a surjective transformation.

Evaluate Definite Integral Using Integral Properties:
If ${\displaystyle {\int }_1^{7}f\left(x\right)dx=2.5}$ and ${\displaystyle {\int }_1^{7}g\left(x\right)dx=4}$, find ${\displaystyle {\int }_1^7[4f\left(x\right)-2g\left(x\right)]dx}$.

Linear transformation with special properties
Linear transformation ${\displaystyle f:R^{10}\to R^7}$ has an attribute that every vector v for which is true that ${\displaystyle f\left(v\right)=0}$ is in linear span ${\displaystyle {(1,2,\dots ,10)}^T,{(1,1\dots ,1)}^T>}$. Create such transformation or prove that it doesn't exists.

Lineal Algebra. Please provide a well explained solution for the following. What is the relationship between matrix transformations and linear transformations? Plus, how to find the matrix that defines a matrix transformation?