Find the matrix form of the linear transformation L(x, y)=( x-4 y -2 x+3 y ) with resp

step 1 For this exercise, we have L of E sub i equals 0. L of E sub 4 is equal to 2 E sub 4. So we have

Find the matrix form of the linear transformation L(x, y)=( ...

Find the matrix form of the linear transformation $L(x, y)=\left(\begin{array}{c}x-4 y \\ -2 x+3 y\end{array}\right)$ with respect to the following bases of $\mathbb{R}^2$ : (a) $\left(\begin{array}{l}1 \\ 0\end{array}\right),\left(\begin{array}{l}0 \\ 1\end{array}\right)$, (b) $\left(\begin{array}{l}2 \\ 0\end{array}\right),\left(\begin{array}{l}0 \\ 3\end{array}\right)$, (c) $\left(\begin{array}{l}1 \\ 1\end{array}\right),\left(\begin{array}{r}-1 \\ 1\end{array}\right)$, (d) $\left(\begin{array}{l}2 \\ 1\end{array}\right)+\left(\begin{array}{r}-1 \\ 1\end{array}\right)$, (e) $\left(\begin{array}{l}3 \\ 2\end{array}\right),\left(\begin{array}{l}2 \\ 3\end{array}\right)$.

Find the matrix form of the linear transformation $L(x, y)=\left(\begin{array}{c}x-4 y \\ -2 x+3 y\end{array}\right)$ with respect to the following bases of $\mathbb{R}^2$ :
(a) $\left(\begin{array}{l}1 \\ 0\end{array}\right),\left(\begin{array}{l}0 \\ 1\end{array}\right)$,
(b) $\left(\begin{array}{l}2 \\ 0\end{array}\right),\left(\begin{array}{l}0 \\ 3\end{array}\right)$,
(c) $\left(\begin{array}{l}1 \\ 1\end{array}\right),\left(\begin{array}{r}-1 \\ 1\end{array}\right)$,
(d) $\left(\begin{array}{l}2 \\ 1\end{array}\right)+\left(\begin{array}{r}-1 \\ 1\end{array}\right)$,
(e) $\left(\begin{array}{l}3 \\ 2\end{array}\right),\left(\begin{array}{l}2 \\ 3\end{array}\right)$.

Key Concepts

Linear Transformation

A linear transformation is a function between vector spaces that preserves addition and scalar multiplication. It is completely determined by its effect on a basis of the domain, and studying its action on these basis vectors allows for a full description of the transformation through a matrix representation.

Basis of a Vector Space

A basis is a set of linearly independent vectors that spans the entire vector space. It provides a coordinate system, meaning any vector in the space can be uniquely expressed as a linear combination of the basis vectors. The choice of basis is pivotal when representing linear transformations as matrices.

Matrix Representation of a Linear Transformation with Respect to Bases

The matrix representation of a linear transformation is obtained by expressing the images of the domain's basis vectors in terms of the codomain's basis. Each column of the resulting matrix is formed from the coordinates of the transformed basis vector. This representation simplifies computations and analysis of the transformation.

Change of Basis

Changing basis involves re-expressing vectors and linear transformations in different coordinate systems. When using non-standard (or alternative) bases, the transformation matrix must be adjusted accordingly, which is done by finding the coordinate representation of transformed vectors relative to the new basis. This concept is crucial for understanding how the representation of a linear map varies with different choices of bases.

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