91影视

A determinant is a unique number associatedwith a square matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

Given linear transformation,

$\mathrm{T}\left(\stackrel{鈫�}{\mathrm{v}}\right)=\left[\begin{array}{l}1\\ 2\\ 3\end{array}\right]脳\stackrel{鈫�}{\mathrm{v}}$

First, we notice that the plane$V$ contains$(0,0,0)$, as well as$(1,1,鈭�1)$ and$(3,0,鈭�1)$. This means that$V$ is a subspace of${\mathrm{鈩�}}^3$, spanned by$(1,1,鈭�1)$ and$(3,0,鈭�1)$.

Now, we compute:

$\mathrm{T}\left(\stackrel{鈫�}{\mathrm{v}}\right)=\left[\begin{array}{l}1\\ 2\\ 3\end{array}\right]脳\stackrel{鈫�}{\mathrm{v}}$

Substitute $\left[\begin{array}{c}1\\ 1\\ 鈭�1\end{array}\right]$ for $\stackrel{鈫�}{v}$ and simplify as:

$\begin{array}{l}T\left(\left[\begin{array}{c}1\\ 1\\ 鈭�1\end{array}\right]\right)=\left[\begin{array}{l}1\\ 2\\ 3\end{array}\right]脳\left[\begin{array}{c}1\\ 1\\ 鈭�1\end{array}\right]\\ T\left(\left[\begin{array}{c}1\\ 1\\ 鈭�1\end{array}\right]\right)=\left[\begin{array}{c}鈭�5\\ 4\\ 鈭�1\end{array}\right]\\ T\left(\left[\begin{array}{c}1\\ 1\\ 鈭�1\end{array}\right]\right)=4\left[\begin{array}{c}1\\ 1\\ 鈭�1\end{array}\right]鈭�3\left[\begin{array}{c}3\\ 0\\ 鈭�1\end{array}\right]\end{array}$

Substitute $\left[\begin{array}{c}3\\ 0\\ 鈭�1\end{array}\right]$ for $\stackrel{鈫�}{v}$ and simplify as:

$\begin{array}{l}T\left(\left[\begin{array}{c}3\\ 0\\ 鈭�1\end{array}\right]\right)=\left[\begin{array}{l}1\\ 2\\ 3\end{array}\right]脳\left[\begin{array}{c}3\\ 0\\ 鈭�1\end{array}\right]\\ T\left(\left[\begin{array}{c}3\\ 0\\ 鈭�1\end{array}\right]\right)=\left[\begin{array}{c}鈭�2\\ 10\\ 鈭�6\end{array}\right]\\ T\left(\left[\begin{array}{c}3\\ 0\\ 鈭�1\end{array}\right]\right)=10\left[\begin{array}{c}1\\ 1\\ 鈭�1\end{array}\right]鈭�4\left[\begin{array}{c}3\\ 0\\ 鈭�1\end{array}\right]\end{array}$

Since$\mathcal{B}=\left\{\right(1,1,鈭�1),(3,0,鈭�1\left)\right\}$is a basis for$V$, this means that the matrix of$\mathrm{T}$corresponding to$\mathcal{B}$is$B=\left[\begin{array}{cc}4& 鈭�3\\ 10& 鈭�4\end{array}\right]$

Therefore, $\mathrm{detT}=\mathrm{detB}=14$.