Show that if T is a linear transformation from a vector space 𝒰 over 𝔽 with basis {𝐮1, …, 𝐮q} in

step 1 By induction, if K is an element of n and v .1, v .2, all the way to v. sub k, this element of r

Show that if T is a linear transformation from a vector...

Show that if $T$ is a linear transformation from a vector space $\mathcal{U}$ over $\mathbb{F}$ with basis $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_q\right\}$ into a vector space $\mathcal{V}$ over $\mathbb{F}$ with basis $\left\{\mathbf{v}_1, \ldots, \mathbf{v}_p\right\}$, then there exists a unique set of scalars $a_{i j} \in \mathbb{F}, i=1, \ldots, p$ and $j=1, \ldots, q$ such that $$ T \mathbf{u}_j=\sum_{i=1}^p a_{i j} \mathbf{v}_i \text { for } j=1, \ldots, q $$ and hence that $$ T\left(\sum_{j=1}^q x_j \mathbf{u}_j\right)=\sum_{i=1}^p y_i \mathbf{v}_i \Longleftrightarrow A \mathbf{x}=\mathbf{y} $$ where $\mathbf{x} \in \mathbb{F}^q$ has components $x_1, \ldots, x_q, \mathbf{y} \in \mathbb{F}^p$ has components $y_1, \ldots, y_p$ and the entries $a_{i j}$ of $A \in \mathbb{F}^{p \times q}$ are determined by formula (1.5).

Show that if $T$ is a linear transformation from a vector space $\mathcal{U}$ over $\mathbb{F}$ with basis $\left\{\mathbf{u}_1, \ldots, \mathbf{u}_q\right\}$ into a vector space $\mathcal{V}$ over $\mathbb{F}$ with basis $\left\{\mathbf{v}_1, \ldots, \mathbf{v}_p\right\}$, then there exists a unique set of scalars $a_{i j} \in \mathbb{F}, i=1, \ldots, p$ and $j=1, \ldots, q$ such that
$$
T \mathbf{u}_j=\sum_{i=1}^p a_{i j} \mathbf{v}_i \text { for } j=1, \ldots, q
$$
and hence that
$$
T\left(\sum_{j=1}^q x_j \mathbf{u}_j\right)=\sum_{i=1}^p y_i \mathbf{v}_i \Longleftrightarrow A \mathbf{x}=\mathbf{y}
$$
where $\mathbf{x} \in \mathbb{F}^q$ has components $x_1, \ldots, x_q, \mathbf{y} \in \mathbb{F}^p$ has components $y_1, \ldots, y_p$ and the entries $a_{i j}$ of $A \in \mathbb{F}^{p \times q}$ are determined by formula (1.5).

Key Concepts

Uniqueness and Existence of Scalars

The existence and uniqueness of the scalars (or matrix entries) that represent a linear transformation stem from the fundamental properties of bases and linearity. Given that every vector has a unique coordinate representation with respect to a basis, the scalars that define the transformation of the basis vectors are uniquely determined. This guarantees a well-defined, one-to-one correspondence between the transformation and its matrix representation.

Matrix Representation of a Linear Transformation

A linear transformation can be represented by a matrix once bases for the domain and codomain have been selected. The matrix entries are determined by how the linear transformation maps the basis vectors of the domain to the codomain. Specifically, each column of the matrix corresponds to the coordinates of the image of a basis vector from the domain expressed in the codomain's basis, creating a concrete representation that can be used for computations.

Linear Transformation

A linear transformation is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. This property ensures that the image of a linear combination of vectors is the corresponding linear combination of their images, making it possible to work within a structured framework for mapping elements between vector spaces.

Basis and Coordinate Representation

A basis of a vector space is a set of linearly independent vectors that spans the space, meaning every vector in the space can be uniquely expressed as a linear combination of the basis vectors. This representation allows every vector to be identified by its coordinates relative to the chosen basis, which is fundamental for many constructions and proofs in linear algebra.

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