- $T:$ $\mathbb{R}^2$ → $\mathbb{R}^4$, $T(e_1)$ $= (3,1,3,1)$ $and$ $T (e_2)$ $= (-5,2,0,0),$ $where$ $e_1$ $= (1,0)$ $and$ $e_2$ $= (0,1)$
In this question, we have to find the standard matrix of the linear transformation $T$.
First, we should recall our concept of the standard matrix. The standard matrix has columns that are the images of the vector of standard basis.
\[A = \left [\begin {matrix}1\\0\\0\\ \end {matrix} \right] B = \left [ \begin {matrix}0\\1\\0\\ \end {matrix}\right] C = \left [ \begin {matrix}0\\0\\1\\ \end {matrix} \right ]\]
The transformation matrix is a matrix that changes the Cartesian system of a vector into a different vector with the help of matrix multiplication.
Expert Answer
Transformation matrix $T$ of order $a \times b$ on multiplication with a vector $X$ of $b$ components represented as a column matrix transforms into another matrix $X’$.
A vector $X= ai + bj$ when multiplied with matrix $T$ $ \left [ \begin {matrix} p&q\\r&s \\ \end {matrix} \right]$ is transformed into another vector $Y=a’i+ bj’$. Thus, a $2 \times 2$ transformation matrix can be shown as below,
\[TX =Y\]
\[ \left[\begin {matrix} p&q\\r&s \\ \end {matrix}\right] \times \left [ \begin {matrix}x\\y\\ \end {matrix} \right] =\left [\begin {matrix}x^\prime\\y^\prime\\ \end {matrix} \right ]\]
There are different types of Transformation matrices such as stretching, rotation, and shearing. It is used in Dot and Cross Product of vectors and can also be used in finding the determinants.
Now applying the above concept on the given question, we know that the standard basis for $R^2$ is
\[e_1=\left [\begin {matrix}1\\0\\ \end {matrix} \right ]\]
and \[e_2= \left [\begin {matrix}1\\0\\ \end {matrix} \right ]\]
and we have
\[T(e_1)= \left [ \begin {matrix}3\\1\\3\\1\\ \end {matrix} \right] T(e_2)= \left [ \begin {matrix}-5\\2\\0\\0\\ \end {matrix} \right ]\]
To find the standard matrix of linear transformation $T$, let us suppose it is matrix $X$ and it can be written as:
\[X = T(e_1) T(e_2)\]
\[X = \left [ \begin {matrix} \begin {matrix}3\\1\\3\\ \end {matrix}& \begin {matrix}-5\\2\\0\\ \end {matrix}\\1&0\\ \end {matrix} \right ]\]
Numerical Results
So the standard matrix for linear transformation $T$ is given as:
\[X =\left [ \begin {matrix} \begin {matrix}3\\1\\3\\ \end {matrix}& \begin {matrix}-5\\2\\0\\ \end {matrix}\\1&0\\ \end {matrix} \right ]\]
Example
Find the new vector formed for the vector $6i+5j$, with the transformation matrix $\left[ \begin {matrix}2&3\\1&-1\\ \end{matrix} \right ]$
Given as:
Transformation matrix \[T = \left [ \begin {matrix}2&3\\1&-1\\ \end {matrix} \right ] \]
Given vector is written as,\[ A = \left [ \begin {matrix}6\\5\\ \end {matrix} \right ] \]
We have to find the transformation matrix B represented as:
\[B = TA\]
Now putting the values in above equation, we get:
\[B=TA=\left [ \begin {matrix}2&3\\1&-1\\\end {matrix} \right ]\times\left [ \begin {matrix}6\\5\\\end {matrix} \right ] \]
\[B=\left [\begin {matrix}2\times6+3\times(5)\\1\times6+(-1)\times5\\\end {matrix} \right ] \]
\[B=\left [\begin {matrix}27\\1\\ \end {matrix} \right ] \]
so based on above matrix, our required transformation standard matrix will be:
\[B = 27i+1j\]