Solving the Matrix Transformation: $B \to C$

In summary: So, in summary, the number $2a_{11}+3a_{12}+a_{21}+3a_{22}$ is equal to the sum of the first and second entries of the first and second columns of the matrix $A$ in basis $C$ which is $\begin{pmatrix}2 & 0 \\ 3 & -1 \end{pmatrix}$, giving us a final result of $2+0+3-1= 4$.
  • #1
evinda
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Hello! (Wave)

Let $B=(b_1, b_2)$, $C=(c_1, c_2)$ basis of $\mathbb{R}^2$ and $L$ operator of $\mathbb{R}^2$, the matrix as for $B$ of which is $\begin{pmatrix}
2 & 2\\
1 & 0
\end{pmatrix}$. If $b_1=c_1+2c_2+b_2=c_1+3c_2$ and $A=\begin{pmatrix}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{pmatrix}$ the matrix of $L$ as for the basis $C$, what does the number $2a_{11}+3a_{12}+a_{21}+3a_{22}$ equal to? (Thinking)

In order to find the desired quantity, do we use the fact that the composition of C with $\begin{pmatrix}
2 & 2\\
1 & 0
\end{pmatrix}$ is equal to $A$ ? But how do we express the composition mathematically? We cannot multiply $C$ by the matrix, since the dimensions do not agree... (Worried)

Or am I somewhere wrong? (Thinking)
 
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  • #2
evinda said:
Hello! (Wave)

Let $B=(b_1, b_2)$, $C=(c_1, c_2)$ basis of $\mathbb{R}^2$ and $L$ operator of $\mathbb{R}^2$, the matrix as for $B$ of which is $\begin{pmatrix}
2 & 2\\
1 & 0
\end{pmatrix}$. If $b_1=c_1+2c_2+b_2=c_1+3c_2$
Do you mean $b_1= c_1+ 3c_2$ and $b_2= c_1+ 3c_2$?

and $A=\begin{pmatrix}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{pmatrix}$ the matrix of $L$ as for the basis $C$, what does the number $2a_{11}+3a_{12}+a_{21}+3a_{22}$ equal to? (Thinking)

In order to find the desired quantity, do we use the fact that the composition of C with $\begin{pmatrix}
2 & 2\\
1 & 0
\end{pmatrix}$ is equal to $A$ ? But how do we express the composition mathematically? We cannot multiply $C$ by the matrix, since the dimensions do not agree... (Worried)

Or am I somewhere wrong? (Thinking)
Perhaps you are confused as to what $C$ is. C is the pair of 2-vectors $c_1$ and $c_2$ so we certainly can multiply each by the 2 by 2 matrix A. Or we can represent C itself as a 2 by 2 matrix with $c_1$ and $c_2$ as columns and multiply that matrix by A.

To write a linear operator in a given ordered basis, apply the linear operator to each base vector in turn and use the result (again written as a linear combination of the basis vectors) as columns.

So how do we apply A to $c_1$? From $b_1= c_1+ 2c_2$ and $b_2= c_1+ 3c_2$, subtracting eliminates $c_1$ and gives $c_2= b_2- b_1$. Then $b_1= c_1+ 2c_2= c_1+ 2(b_2- b_1)= c_1+ 2b_2- 2b_1$ so $c_1= b_1- 2b_2+ 2b_1= 3b_1- 2b_2$.

So $Ac_1= A(3b_1- 2b_2)= 3Ab_1- 2Ab_2$. But $Ab_1$ is just the first column of the matrix form for A in basis B: $Ab_1= \begin{pmatrix}2 \\ 1 \end{pmatrix}$. Similarly, $Ab_2$ is the second column: $Ab_2= \begin{pmatrix}2 \\ 0 \end{pmatrix}$. Then $Ac_1= 3\begin{pmatrix}2 \\ 1\end{pmatrix}- 2\begin{pmatrix}2 \\ 0 \end{pmatrix}= \begin{pmatrix}2 \\ 3\end{pmatrix}$.

And $Ac_2= A(b_2- b_1)= Ab_2- Ab_1= \begin{pmatrix}2 \\ 0 \end{pmatrix}- \begin{pmatrix}2 \\ 1 \end{pmatrix}= \begin{pmatrix}0 \\ -1\end{pmatrix}$.

Finally, linear operator A, represented as a matrix in basis C, is the matrix having those vectors as columns:
$\begin{pmatrix}2 & 0 \\ 3 & -1 \end{pmatrix}$.
 

FAQ: Solving the Matrix Transformation: $B \to C$

What is a matrix transformation?

A matrix transformation is a mathematical process in which a matrix (a rectangular array of numbers) is used to transform one set of data into another set of data. It involves multiplying the matrix by a vector or another matrix to produce a new set of data values.

Why is solving the matrix transformation important?

Solving the matrix transformation is important because it allows us to manipulate and analyze data in a more efficient and organized manner. It is commonly used in fields such as computer graphics, physics, economics, and engineering.

What are the steps to solve a matrix transformation?

The steps to solve a matrix transformation are as follows:

  • Identify the given matrix and its dimensions
  • Determine the transformation matrix (the matrix used to transform the given matrix)
  • Perform the transformation by multiplying the given matrix by the transformation matrix
  • Identify the resulting matrix and its dimensions

What are some common types of matrix transformations?

Some common types of matrix transformations include scaling, rotation, shearing, and reflection. These transformations can be applied to both 2-dimensional and 3-dimensional data.

What are some real-world applications of matrix transformations?

Matrix transformations have a wide range of real-world applications. They are used in computer graphics to create 3D images and animations, in physics to model physical systems, in economics to analyze data and make predictions, and in engineering to design structures and systems. They are also used in machine learning and data analysis to process and manipulate large datasets.

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