Matrix Transformations between dimensions

[ChaoticLlama]

I missed the lecture on the topic of matrix transformations, and my textbook does not sufficiently help me. Are there any online tutorials that you would recommend?

Specifically I am having difficulties with transformations between dimensions, (ie: \(\displaystyle \L\ T:\mathbb{R}^n \to \mathbb{R}^m \\)).
In my book it states that for a transformation \(\displaystyle \L\ T:\mathbb{R}^4 \to \mathbb{R}^3 \\) is defined by the equation:
\(\displaystyle \L\
T\left[ {\begin{}}
{x_1 } \\
{x_2 } \\
{x_3 } \\
{x_4 } \\
\\right] = \left[ {\begin{}}
{x_1 + x_2 } \\
{x_2 + x_3 } \\
{x_3 + x_4 } \\
\\right]\)

Is this correct? I've found conflicting information online.

Thanks for your help.

 

[stapel]

Is this correct?

Are you asking whether it is correct that T represents a linear transformation...? (There are infinitely-many such transformations, of course.)

Thank you.

Eliz.

 

[pka]

Yes. In the case you gave the transformation would be:
\(\displaystyle \L
T = \left[ {\begin{array}{cccc}
1 & 1 & 0 & 0 \\
0 & 1 & 1 & 0 \\
0 & 0 & 1 & 1 \\
\end{array}} \right].\)

 

[ChaoticLlama]

Ah, that makes a little more sense now. The transformation I stated above was done through a specific matrix.

So in general, to transform a 4-D matrix X with dimensions 4xm one would multiply by an arbitrary matrix A with dimensions 3x4 to yield the resultant 3-D matrix Y of dimenions 3xm. [AX=Y]

Does the above statement make sense?

 

[pka]

So in general, to transform a 4-D matrix X with dimensions 4xm one would multiply by an arbitrary matrix A with dimensions 3x4 to yield the resultant 3-D matrix Y of dimenions 3xm. [AX=Y]

The correct statement would be to transform a 4-dimensional (column) vector to 3-dimensions.
This is done by a linear transformation. Linear transformations have matrix representations.
In this case the matrix is a 3x4 with output 3x1 column vector in R<SUP>3</SUP>.

 

[ChaoticLlama]

ahhh I see. thank you.