# 072 is Q(theta) a linear transformation from R^2 to itself.

• MHB
• karush
In summary: Better is to understand what "linear transformation" means! ANY transformation that can be written as a matrix multiplication is linear! A transformation, L, on a vector space is "linear" if and only ifL(u+ v)= Lu+ Lv, for any vectors u and v, andL(au)= aLu, for any vector u and scalar, a.Here if $u= \begin{pmatrix}x \\ y \end{pmatrix}$ and $v= \begin{pmatrix} a \\ b\end{pmatrix}$, $L(u+ v)= \begin{pm karush Gold Member MHB if$Q(\theta)$is$\left[\begin{array}{rr}
\cos{\theta}&- \sin{\theta}\\
\sin{\theta}&\cos{\theta}
\end{array}\right]$how is$Q(\theta)$is a linear transformation from R^2 to itself. ok I really didn't know a proper answer to this question but presume we would need to look at the unit circle not sure if this helps Last edited by a moderator: Ok thanks I usually don't get much replies on these linear algebra posts karush said: Ok thanks I usually don't get much replies on these linear algebra posts I'd help out, but it's been since my sophomore year in school (1973) since I've taken a course in Linear Algebra ... except for the very basic stuff, I haven't used it so I've "losed" it. Better is to understand what "linear transformation" means! ANY transformation that can be written as a matrix multiplication is linear! A transformation, L, on a vector space is "linear" if and only if L(u+ v)= Lu+ Lv, for any vectors u and v, and L(au)= aLu, for any vector u and scalar, a. Here if$u= \begin{pmatrix}x \\ y \end{pmatrix}$and$v= \begin{pmatrix} a \\ b\end{pmatrix}$,$L(u+ v)= \begin{pmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{pmatrix}\begin{pmatrix}x+ a \\ y+ b\end{pmatrix}= \begin{pmatrix}(x+ a)cos(\theta)- (y+ b)sin(\theta) \\ (x+ a)sin(\theta)+ (y+ b)cos(\theta)\end{pmatrix}$. While$Lu+ Lv= \begin{pmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{pmatrix}\begin{pmatrix}x \\ y \end{pmatrix}+ \begin{pmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{pmatrix}\begin{pmatrix} a \\ b \end{pmatrix}= \begin{pmatrix} xcos(\theta)- ysin(\theta) \\ xsin(\theta)+ y cos(\theta)\end{pmatrix}+ \begin{pmatrix} acos(\theta)- bsin(\theta) \\ asin(\theta)+ bcos(\theta)\end{pmatrix}= \begin{pmatrix}(x+ a)cos(\theta)- (y+ b)sin(\theta) \\ (x+ a)sin(\theta)+ (y+ b)cos(\theta)\end{pmatrix}$. And$L(au)= \begin{pmatrix} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta)\end{pmatrix}\begin{pmatri
072 is Q(theta) is a linear transformation from R^2 to itself.
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karush
Well-known member

Jan 31, 2012 2,838
if Q(θ)Q(θ) is

[cosθsinθ−sinθcosθ][cos⁡θ−sin⁡θsin⁡θcos⁡θ]

how is Q(θ)Q(θ) is a linear transformation from R^2 to itself.

ok I really didn't know a proper answer to this question but presume we would need to look at the unit circle

not sure if this helps

Last edited by a moderator: Today at 12:07 AM
https://mathhelpboards.com/posts/124753/report

x}ax \\ ay \end{pmatrix}= \begin{pmatrix}ax coz

skeeter said:
I'd help out, but it's been since my sophomore year in school (1973) since I've taken a course in Linear Algebra ... except for the very basic stuff, I haven't used it so I've "losed" it.

wow... my senior year was 1970 but my highest level in math was algebra II which today is much more advanced

## 1. What does Q(theta) represent in this context?

Q(theta) represents a linear transformation from R^2 to itself, meaning it takes a vector from R^2 and maps it to another vector in R^2 using a linear function.

## 2. How is Q(theta) related to theta?

Theta represents the parameters or coefficients of the linear transformation Q(theta). These parameters determine the specific mapping from one vector to another.

## 3. Can Q(theta) be represented by a matrix?

Yes, Q(theta) can be represented by a 2x2 matrix. The coefficients of the linear transformation would correspond to the entries in the matrix.

## 4. What is the significance of Q(theta) being a linear transformation?

As a linear transformation, Q(theta) preserves vector addition and scalar multiplication, making it a useful tool in many mathematical and scientific applications.

## 5. How is Q(theta) different from other types of transformations?

Unlike other types of transformations, such as nonlinear or affine transformations, Q(theta) is a linear transformation, meaning it follows the properties of linearity and can be represented by a matrix.

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