Assume that T defines a linear transformation and use the...

Assume that $T$ defines a linear transformation and use the given information to find the matrix
of $T.$
$T: \mathbb{R}^{2} \rightarrow \mathbb{R}^{4}$ such that $T(-1,1)=(1,0,-2,2)$ and $T(1,2)=(-3,1,1,1).$

Key Concepts

Linear Transformation

A linear transformation is a function between vector spaces that preserves vector addition and scalar multiplication. This means that the image of a sum of vectors is equal to the sum of the images of the vectors, and the image of a scalar multiple of a vector is equal to that scalar multiplied by the image of the vector.

Matrix Representation of a Linear Transformation

Every linear transformation between finite-dimensional vector spaces can be represented by a matrix. This matrix is constructed by applying the transformation to each vector in a chosen basis of the domain and arranging the resulting images, expressed in terms of a basis of the codomain, as columns of the matrix.

Basis and Coordinate Representation

A basis is a set of linearly independent vectors that span the entire vector space. When both the domain and codomain have specified bases, any vector can be uniquely expressed in terms of these bases, making it possible to represent linear transformations as matrices that convert coordinate vectors from the domain to the codomain.

Determining a Matrix from Given Transformations

If the images of a set of basis vectors under a linear transformation are known, the corresponding matrix of the transformation can be determined by expressing these images as coordinate vectors relative to the codomain basis. This process involves aligning each coordinate vector as a column in the matrix, thereby fully capturing the action of the transformation.

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