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

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
  • #1
karush
Gold Member
MHB
3,269
5
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

Screenshot 2021-03-13 1.27.12 PM.png
 
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  • #2
Ok thanks
I usually don't get much replies on these linear algebra posts
 
  • #3
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.
 
  • #4
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
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072 is Q(theta) is a linear transformation from R^2 to itself.
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karush
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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
screenshot-2021-03-13-1-27-12-pm-png.gif


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x}ax \\ ay \end{pmatrix}= \begin{pmatrix}ax coz
 
  • #5
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
2021_03_07_16.48.50.jpg
 

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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