Question

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

Transformation properties
ANSWERED
asked 2020-12-25

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 \(A=\begin{bmatrix}a & b \\c & d \end{bmatrix}\)

Answers (1)

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

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