Let T: ℝ^3 →ℝ^2 and S: ℝ^2 →ℝ^3 be the linear transformations given by T(x1, x2, x3)=(x1+x2, x2+

step 1 In this problem, we have been given two linear transformations. And what we need to determine is

Let T: ℝ^3 →ℝ^2 and S: ℝ^2 →ℝ^3 be the linear...

Let $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ and $S: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ be the linear transformations given by $$ T\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}+x_{2}, x_{2}+x_{3}\right) $$ and $$ S\left(x_{1}, x_{2}\right)=\left(x_{1}+x_{2}, x_{1}-x_{2}, x_{2}\right) . $$ Give similar formulae for $S T$ and $T S$.

Let $T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ and $S: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$ be the linear transformations given by
$$
T\left(x_{1}, x_{2}, x_{3}\right)=\left(x_{1}+x_{2}, x_{2}+x_{3}\right)
$$
and
$$
S\left(x_{1}, x_{2}\right)=\left(x_{1}+x_{2}, x_{1}-x_{2}, x_{2}\right) .
$$
Give similar formulae for $S T$ and $T S$.

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 linear combination of vectors is the same as the corresponding linear combination of the images. Understanding linear transformations is fundamental in linear algebra as they provide a framework for solving and analyzing systems of linear equations, changing coordinates, and many other applications.

Composition of Linear Transformations

The composition of linear transformations involves applying one transformation after another. If you have two linear transformations, say T and S, then the composition S ? T is defined by applying T to a vector first and then applying S to the result. This concept is essential because it shows that the combination of linear operations is itself linear, and it allows for the construction of complex mappings from simpler components.

Matrix Representation of Linear Transformations

Every linear transformation can be represented by a matrix once bases for the domain and codomain are chosen. When transformations are composed, their corresponding matrices are multiplied to yield the matrix of the composite transformation. This representation is crucial as it converts functional operations into matrix operations, enabling the use of algebraic and computational techniques to analyze and compute the effects of successive linear transformations.

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