3. Linear transformations and matrices

Linear transformations and matrices

What to learn

• What is the Linear Transformation

  Description

• Transformation

  • Take some input vector to output vector - = input vector moving over output vector - We will treat Only Linear Transformation In Linear Algebra - Special case of Transformation - Lines remain lines - Origin remains fixed - 2d Linear Transformation is fully discribed by two basis vectors - Always same linear combination about two basis vectors - And if we can explain 2d Linear Transformation using Matrix, and we can find out where a specific vector will move. - Example

                    If we make 2d linear transformation that move i hat to [3,2]^T and j hat to [2,1]^T. and wonder where [5,7]^T will move.
    
                    $$\vec{i} = \begin{bmatrix}1\\ 0\end{bmatrix},\vec{j} = \begin{bmatrix}0\\ 1\end{bmatrix} \rightarrow \vec{i} = \begin{bmatrix}3\\ 2\end{bmatrix},\vec{j} = \begin{bmatrix}2\\ 1\end{bmatrix}$$
    
                    $$5 * \begin{bmatrix}3\\ 2\end{bmatrix} + 7 \begin{bmatrix}2\\ 1\end{bmatrix} = 
                    \begin{bmatrix}29\\ 17\end{bmatrix}$$
    
                    [29,17]^T will be the answer. Another way, we can make matrix using i hat and j hat
    
                    $$\begin{bmatrix}3\\ 2\end{bmatrix}\begin{bmatrix}2\\ 1\end{bmatrix}\rightarrow \begin{bmatrix}3 & 2\\ 2 & 1\end{bmatrix}$$
    
                    Then we can get result using matrix calculation.
    
                    $$\begin{bmatrix}3 & 2\\ 2 & 1\end{bmatrix}\begin{bmatrix}5\\ 7\end{bmatrix} = 
                    \begin{bmatrix}29\\ 17\end{bmatrix}$$
    
                    This way, we can get same result.
    
            - If matrix is compounded of Linearly dependent columns, then it makes all vectors move to the vectors' span. ## Next Step
    

• Matrix multiplication as composition

  References

• https://www.youtube.com/watch?v=kYB8IZa5AuE&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=3