Using the last exercise as a guide, justify the following...

Using the last exercise as a guide, justify the following statement Let $\vec{v}_{1}, \vec{v}_{2}, \ldots, \vec{v}_{m}$ be vectors in $\mathbb{R}^{m}$ such that the $\mathrm{ma}$ trix $$S=\left[\begin{array}{cccc} | & | & & | \\ \vec{v}_{1} & \vec{v}_{2} & \dots & \vec{v}_{m} \\ | & | & & | \end{array}\right]$$ is invertible. Let $\vec{w}_{1}, \vec{w}_{2}, \ldots, \vec{w}_{m}$ be arbitrary vectors in $\mathbb{R}^{n} .$ Then there exists a unique linear transformation $T$ from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$ such that $T\left(\vec{v}_{i}\right)=\vec{w}_{i},$ for all $i=1, \ldots, m,$ Find the matrix $A$ of this transformation in terms of $S$ and \[\boldsymbol{B}=\left[\begin{array}{cccc} | & | & & | \\ \vec{w}_{1} & \vec{w}_{2} & \dots & \vec{w}_{m} \\ | & | & & | \end{array}\right].\]

Using the last exercise as a guide, justify the following statement Let $\vec{v}_{1}, \vec{v}_{2}, \ldots, \vec{v}_{m}$ be vectors in $\mathbb{R}^{m}$ such that the $\mathrm{ma}$ trix
$$S=\left[\begin{array}{cccc}
| & | & & | \\
\vec{v}_{1} & \vec{v}_{2} & \dots & \vec{v}_{m} \\
| & | & & |
\end{array}\right]$$
is invertible. Let $\vec{w}_{1}, \vec{w}_{2}, \ldots, \vec{w}_{m}$ be arbitrary vectors in $\mathbb{R}^{n} .$ Then there exists a unique linear transformation $T$ from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$ such that $T\left(\vec{v}_{i}\right)=\vec{w}_{i},$ for all $i=1, \ldots, m,$ Find the matrix $A$ of this transformation
in terms of $S$ and
\[\boldsymbol{B}=\left[\begin{array}{cccc}
| & | & & | \\
\vec{w}_{1} & \vec{w}_{2} & \dots & \vec{w}_{m} \\
| & | & & |
\end{array}\right].\]

Key Concepts

Linear Transformation

A linear transformation is a function between two vector spaces that preserves vector addition and scalar multiplication. Its behavior is completely determined when you know its effect on a basis of the domain, making it possible to extend that action to the entire space.

Invertible Matrix

An invertible matrix (or nonsingular matrix) represents a bijective linear transformation from a vector space to itself. Its invertibility guarantees that every element of the domain is uniquely mapped to the codomain, and it is essential in constructing the inverse of a coordinate mapping.

Basis and Coordinate Representation

A basis is a collection of linearly independent vectors that span a vector space. Knowing the coordinates of vectors with respect to a basis allows any vector in the space to be expressed uniquely, which is critical in both defining and manipulating linear transformations.

Matrix Representation of a Linear Transformation

The matrix representation of a linear transformation is formed by expressing the images of the basis vectors of the domain in terms of the codomain's coordinates. This matrix provides a systematic way for computing the transformation on any vector via matrix multiplication.

Uniqueness of the Linear Transformation

When a linear transformation is defined by its action on a basis, the transformation is uniquely determined because every vector in the domain has a unique linear combination of the basis vectors. This ensures that the matrix representing the transformation is the only one that satisfies the mapping conditions.

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