Show that T is a linear transformation.

Question

Let B be an element of $R^{n \times n}$ and define $T(A) = BAB$ for all $A \in R^{n \times n}$. Show that T is a linear transformation.

I am completely lost and I do not know how to start this.

Answer 1

Here $ T(aA)=BaAB=aBAB=aT (A)$ $\forall a\in \mathbb { R } $

And $T(A+C)=B(A+C)B=BAB+BCB=T(A)+T(C)$ $\forall A,C\in \mathbb {R}^{n\times n}$

Thus $T$ is linear.

Answer 2

This is easy if one works from the axiomatic definition of linearity. Recall that $T$ is a linear transformation if and only if

$T(\alpha X + \beta Y) = \alpha T(X) + \beta T(Y), \tag{1}$

for all $\alpha, \beta \in R$ and all $X, Y \in R^{n \times n}$. So we have

$T(\alpha X + \beta Y) = B(\alpha X + \beta Y)B = B(\alpha X)B + B(\beta Y)B = \alpha BXB + \beta BYB = \alpha T(X) + \beta T(Y); \tag{2}$

$T$ is thus a linear transformation operating on $R^{n \times n}$. QED.

Hope this helps. Cheerio,

and as always,

Fiat Lux!!!