uplakanimkk

2022-07-14

Suppose $T$ is a transformation from ${\mathbb{R}}^{2}$ to ${\mathbb{R}}^{2}$. Find the matrix $A$ that induces $T$ if $T$ is the (counter-clockwise) rotation by $\frac{3}{4}}\pi $.

how to begin to find a matrix that is 2x2 for this question.

how to begin to find a matrix that is 2x2 for this question.

Dayana Zuniga

Beginner2022-07-15Added 16 answers

The rotation matrix is $\left[\begin{array}{cc}\mathrm{cos}(\theta )& -\mathrm{sin}(\theta )\\ \mathrm{sin}(\theta )& \mathrm{cos}(\theta )\end{array}\right]$But blindly applying this formula doesn't teach you the most important part of linear transformations. (ANY linear transformation matrix is defined by where it takes the unit vectors.)

This linear transformation T that is a counterclockwise rotation takes $\left[\begin{array}{c}1\\ 0\end{array}\right]$ to $\left[\begin{array}{c}\mathrm{cos}(\theta )\\ \mathrm{sin}(\theta )\end{array}\right]$ and $\left[\begin{array}{c}0\\ 1\end{array}\right]$ to $\left[\begin{array}{c}-\mathrm{sin}(\theta )\\ \mathrm{cos}(\theta )\end{array}\right]$. You can confirm this on your own using geometry.

And this works for ANY linear transformation.

This linear transformation T that is a counterclockwise rotation takes $\left[\begin{array}{c}1\\ 0\end{array}\right]$ to $\left[\begin{array}{c}\mathrm{cos}(\theta )\\ \mathrm{sin}(\theta )\end{array}\right]$ and $\left[\begin{array}{c}0\\ 1\end{array}\right]$ to $\left[\begin{array}{c}-\mathrm{sin}(\theta )\\ \mathrm{cos}(\theta )\end{array}\right]$. You can confirm this on your own using geometry.

And this works for ANY linear transformation.

An object moving in the xy-plane is acted on by a conservative force described by the potential energy function

where$U(x,y)=\alpha (\frac{1}{{x}^{2}}+\frac{1}{{y}^{2}})$ is a positive constant. Derivative an expression for the force expressed terms of the unit vectors$\alpha$ and$\overrightarrow{i}$ .$\overrightarrow{j}$ I need to find a unique description of Nul A, namely by listing the vectors that measure the null space

?

$A=\left[\begin{array}{ccccc}1& 5& -4& -3& 1\\ 0& 1& -2& 1& 0\\ 0& 0& 0& 0& 0\end{array}\right]$T must be a linear transformation, we assume. Can u find the T standard matrix.$T:{\mathbb{R}}^{2}\to {\mathbb{R}}^{4},T\left({e}_{1}\right)=(3,1,3,1)\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}T\left({e}_{2}\right)=(-5,2,0,0),\text{}where\text{}{e}_{1}=(1,0)\text{}{\textstyle \phantom{\rule{1em}{0ex}}}\text{and}{\textstyle \phantom{\rule{1em}{0ex}}}\text{}{e}_{2}=(0,1)$

?Find a nonzero vector orthogonal to the plane through the points P, Q, and R. and area of the triangle PQR

Consider the points below

P(1,0,1) , Q(-2,1,4) , R(7,2,7).

a) Find a nonzero vector orthogonal to the plane through the points P,Q and R.

b) Find the area of the triangle PQR.Consider two vectors A=3i - 1j and B = - i - 5j, how do you calculate A - B?

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A)Weight;

B)Nuclear spin;

C)Momentum;

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