Calculation:

For the vector space V, that contains all \(\displaystyle{2}\times{2}\) matrices

Consider two matrices be lie in that vector space V be,

\(A=\begin{bmatrix}a & b \\c & d \end{bmatrix} B=\begin{bmatrix}p & q \\r & s \end{bmatrix}\) and \(\)

Such that

\(\displaystyle{T}{\left({A}\right)}={a}\ +\ {2}{b}\ -\ {c}\ +\ {d}\)

\(\displaystyle{T}{\left({B}\right)}={p}\ +\ {2}{q}\ -\ {r}\ +\ {s}\)

Perform additional operation.

\(A\ +\ B=\begin{bmatrix}a & b \\c & d \end{bmatrix}+\begin{bmatrix}p & q \\r & s \end{bmatrix}\)

\(=\begin{bmatrix}a + p + b + q \\c + r + d + s \end{bmatrix}\)

And,

\(\displaystyle{T}{\left({A}\right)}\ +\ {T}{\left({B}\right)}={\left({a}\ +\ {2}{b}\ -\ {c}\ +\ {d}\right)}\ +\ {\left({p}\ +\ {2}{q}\ -\ {r}\ +\ {s}\right)}\)

\(\displaystyle={a}\ +\ {p}\ +\ {2}{b}\ +\ {2}{q}\ -\ {c}\ -\ {r}\ +\ {d}\ +\ {s}\)

\(\displaystyle={\left({a}\ +\ {p}\right)}\ +\ {2}{\left({b}\ +\ {q}\right)}\ -\ {\left({c}\ +\ {r}\right)}\ +\ {\left({d}\ +\ {s}\right)}\)

Also, \(\displaystyle{T}{\left({A}\ +\ {B}\right)}={\left({a}\ +\ {p}\right)}\ +\ {2}{\left({b}\ +\ {q}\right)}\ -\ {\left({c}\ +\ {r}\right)}\ +\ {\left({d}\ +\ {s}\right)}\)

Thus it shows that \(\displaystyle{T}{\left({A}+{B}\right)}={T}{\left({A}\right)}+{T}{\left({B}\right)}\), that means it satisfies the addition property

Consider the condition of

\(\displaystyle{T}{\left({k}{A}\right)}={k}{a}\ +\ {2}{k}{b}\ -\ {k}{c}\ +\ {k}{d}\)

\(\displaystyle={k}{\left({a}\ +\ {2}{b}\ -\ {c}\ +\ {d}\right)}\)

\(\displaystyle={k}{T}{\left({A}\right)}\)

So the following expression also satisfies the scalar property.

Answer:

As for the vector space V, the property of addition and scalar are satisfied. Thus the expression of \(\displaystyle{T}{\left({A}\right)}={a}+{2}{b}-{c}+{d}\) is the linear transformation