Question

# For the V vector space contains all 2 times 2 matrices.

Transformation properties

For the V vector space contains all $$\displaystyle{2}\times{2}$$ matrices. Determine whether the $$\displaystyle{T}:{V}\rightarrow{R}^{{1}}$$ is the linear transformation over the $$\displaystyle{T}{\left({A}\right)}={a}\ +\ {2}{b}\ -\ {c}\ +\ {d},$$ where $$A=\begin{bmatrix}a & b \\c & d \end{bmatrix}$$

2020-12-26

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
$$\displaystyle{T}{\left({A}\right)}={a}\ +\ {2}{b}\ -\ {c}\ +\ {d}$$
$$\displaystyle{T}{\left({B}\right)}={p}\ +\ {2}{q}\ -\ {r}\ +\ {s}$$
$$A\ +\ B=\begin{bmatrix}a & b \\c & d \end{bmatrix}+\begin{bmatrix}p & q \\r & s \end{bmatrix}$$
$$=\begin{bmatrix}a + p + b + q \\c + r + d + s \end{bmatrix}$$
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.
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