Chapter 7.CM, Problem 1CM

To determine

To prove:

If $T:R^3\to R^2$, $T\left(x,y,z\right)=\left(2x,x+y\right)$ is a linear transformation.

Answer to Problem 1CM

Solution:

The given function $T$ is a linear transformation.

Explanation of Solution

Given:

The function $T:R^3\to R^2$, $T\left(x,y,z\right)=\left(2x,x+y\right)$.

Approach:

A function $T:A\to B$ is a linear transformation if for any two vectors, $a,b\in A$ and any scalar $c$, the following two properties hold:

1) $T\left(a+b\right)=T\left(a\right)+T\left(b\right)$

2) $T\left(ca\right)=cT\left(a\right)$

Calculation:

Consider any two elements, $a,b\in R^3$.

Then, $a$ and $b$ are of the form:

$\begin{array}{l}a=\left(a_1,a_2,a_3\right)\\ b=\left(b_1,b_2,b_3\right)\end{array}$

Consider,

$\begin{array}{rll}T\left(a+b\right)& =& T\left(a_1+b_1,a_2+b_2,a_3+b_3\right)\\ & =& \left(2\left(a_1+b_1\right),a_1+b_1+a_2+b_2\right)\\ & =& \left(2a_1,a_1+a_2\right)+\left(2b_1,b_1+b_2\right)\\ & =& T\left(a\right)+T\left(b\right)\end{array}$

So, property 1 is true for $T$.

Now, consider,

$\begin{array}{rll}T\left(ca\right)& =& T\left(c{a}_1,c{a}_2,c{a}_3\right)\\ & =& \left(2\left(c{a}_1\right),c{a}_1+c{a}_2\right)\\ & =& c\left(2a_1,a_1+a_2\right)\\ & =& cT\left(a\right)\end{array}$

So, property 2 is true for $T$.

Since, both the properties of a linear transformation are satisfied by $T$, it is a linear transformation.

Therefore, the given function $T$ is a linear transformation.

Conclusion:

Hence, the given function $T$ is a linear transformation.