Find a linear function L: ℝ^2 →ℝ^2 such that L( 1 2 )=( 2 -1 ) and L( 2

step 1 Okay, in this question, we want to find the line L that passes through the points V1 and V2. So

Find a linear function L: ℝ^2 →ℝ^2 such that L( 1 ...

Find a linear function $L: \mathbb{R}^2 \rightarrow \mathbb{R}^2$ such that $L\left(\begin{array}{l}1 \\ 2\end{array}\right)=\left(\begin{array}{r}2 \\ -1\end{array}\right)$ and $L\left(\begin{array}{l}2 \\ 1\end{array}\right)=\left(\begin{array}{r}0 \\ -1\end{array}\right)$.

Key Concepts

Linear Transformations

A linear transformation is a function between two vector spaces that preserves both vector addition and scalar multiplication. This means that the transformation of a linear combination of vectors is the same as the corresponding linear combination of their images. Understanding this key property is essential in determining the mapping by defining the image of basis vectors, from which the transformation of any vector can be constructed.

Matrix Representation

Any linear transformation from ?² to ?² can be represented as a 2×2 matrix. This matrix acts on coordinate vectors via multiplication, providing a systematic way to compute the image of any vector under the transformation. The entries of this matrix are determined by how the transformation maps basis vectors.

Basis and Coordinate Vectors

A basis of a vector space is a set of vectors that are linearly independent and span the entire space. When a linear transformation is defined by its action on basis vectors, it can be extended to any vector in the space by expressing that vector as a linear combination of the basis vectors. This concept is fundamental in constructing and understanding linear maps.

Solving Systems of Linear Equations

Determining the coefficients of a matrix representation for a linear transformation often involves setting up and solving a system of linear equations. These equations are obtained by equating the transformed images of specific vectors (often basis vectors) to their given outputs. Solving these systems is crucial for finding the explicit form of the transformation matrix.

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