Find the Standard Matrix For each linear transformation T: ℝ^n →ℝ^m in Problems 33-40, determine

step 1 In this example, we have a transformation T that is going to be linear, and what it does is it t

Find the Standard Matrix For each linear transformation T:...

Find the Standard Matrix For each linear transformation $T: \mathbb{R}^n \rightarrow \mathbb{R}^m$ in Problems 33-40, determine the standard matrix $\mathbf{A}$ such that $T(\overrightarrow{\mathbf{v}})=\mathbf{A} \overrightarrow{\mathbf{v}}$.
$T\left(v_1, v_2, v_3\right)=v_1+v_3$

Key Concepts

Matrix Multiplication

Matrix multiplication is the operation used to apply the linear transformation to any vector by multiplying the standard matrix by the vector. This operation corresponds to performing a weighted combination of the transformed basis vectors, resulting in the overall transformation of the input vector.

Basis Vectors

Basis vectors are a set of linearly independent vectors that span the vector space. In the context of the standard matrix, the standard basis vectors are used to determine how the linear transformation acts on each axis direction. The transformation's output for each basis vector defines the columns of the standard matrix.

Linear Transformation

A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. This property allows the transformation to be completely described by its effects on a basis of the domain, making it possible to express the transformation using matrix multiplication.

Standard Matrix Representation

The standard matrix representation of a linear transformation is a matrix that, when multiplied by a vector from the domain, yields the corresponding output vector in the codomain. This matrix is constructed by applying the transformation to the standard basis vectors and using the resulting vectors as the columns of the matrix.

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