Linear Transformation Properties: Expert Help & Solutions

Answer true or false to the following statement and explain your answer.
A power transformation with an exponent of zero is the logarithm transformation.

Linear Algebra - Isomorphism, Matrix of Linear Transformation
Let ${\displaystyle T:R^n\to R^n}$ be a linear transformation. Let A be the standard matrix for T . Let ${\displaystyle \beta }$ be an ordered basis for ${\displaystyle R^n}$ and B is a matrix whose columns are vectors in ${\displaystyle \beta }$. Prove${\displaystyle {\left[T\right]}_{\beta }=B^{-1}AB}$.

Which one is correct?
Suppose V is an inner product space whith dim V=7 and U is a subspace of V with dim U=2. Then dim ${\displaystyle U^{\mathrm{\perp }}}$=
A.5.
B.7.
C.9.
D.1.
E.2.

Which one is the correct transformation?
Rewrite the function below from the given standard form into the transformational form of ${\displaystyle f\left(x\right)=q\left(x\right)+\frac{r\left(x\right)}{d\left(x\right)}}$ in order to identify the transformations.
Given Standard form: ${\displaystyle f\left(x\right)=\frac{5x-9}{x-3}}$
Dilation by a factor of 6 with respect to the x-axis, Translation right 3 units and up 5 units
Dilation by a factor of 6 with respect to the x-axis, Translation right 5 units and down 3 units
Dilation by a factor of 1/6 with respect to the y-axis, Translation right 5 units and up 3 units
Dilation by a factor of 1/6 with respect to the y-axis, Translation right 3 units and up 5 units
Dilation by a factor of 5 with respect to the x-axis, Translation left 3 units and up 6 units

Can the function g be obtained from f by transformations? If so, describe the transformations needed?
The function f and g are described algebrically as follows:
${\displaystyle f\left(x\right)={(x+2)}^2}$
${\displaystyle g\left(x\right)={(x-2)}^2+5}$

Let ${\displaystyle T:R^2\to R^2}$ such that T(v) = Av + b, where A is a ${\displaystyle 2\to 2}$ matrix. (Such a transformation is called an affine transformation.) Prove that T is a linear transformation if and only if b = 0.

Give a counterexample to show that the given transformation is not a linear transformation.
$T\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{c}xy\\ x+y\end{array}\right]T$

Apply the transformation M to the polygon with the given vertices. Identify and describe the transformation. ${\displaystyle M:(x,y)\to (–x,–y)}$

Mark each of the following statement true or false:
The transformation ${\displaystyle T:R_n\to R}$ defined by T(x) = | x | is a linear transformation.

Trying to go ABC is reflected across the X axis and then across the Y access which rotation is equivalent to this composition transformations transformations.

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?

The following are each a result of a single transformation. What type of transformation occurred for image F'.
$\begin{array}{|cc|}\hline Pre-Image& Image\\ A(3,4)& A^{\prime }(-4,-5)\\ B(5,10)& B^{\prime }(15,30)\\ C(2,-3)& C^{\prime }(-2,-3)\\ D(3,7)& D^{\prime }(-7,3)\\ E(2,6)& E^{\prime }(5,0)\\ F(5,9)& F^{\prime }(5,-9)\\ \hline\end{array}$

What do you mean by the Karl Pearsons’ coefficient of correlation? What are the properties of the coefficient of correlation?

Define the term Linear Transformation.

If ${\displaystyle \alpha \text{and}\beta }$ are transformations on a set S, prove that ${\displaystyle {\alpha }^{-1}\circ {\beta }^{-1}}$ and ${\displaystyle {\beta }^{-1}\circ {\alpha }^{-1}}$ are transformations. Which is the inverse of ${\displaystyle \alpha \circ \beta }$ ? Prove your answer.

Designer functions Design a sine function with the given properties.
It has a period of 12 hr with a minimum value of ${\displaystyle -4}$ at ${\displaystyle t=0}$ hr and a maximum value of 4 at ${\displaystyle t=6}$ hr.

Let ${\displaystyle L:V\to W}$ be a linear transformation. Explain the meaning of the following statement: The action of the linear transformation L is completely determined by its action on a basis for V.

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.

Explain Eigenvector of matrix transformation?