Category: Don Shimamoto’s Multivariable Calculus (2020)

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202109101713 Exercises 1.2.1

Let T:\mathbb{R}^2\rightarrow \mathbb{R}^3 be the function defined by T(x_1,x_2)=(x_1,x_2,0). Show that T is a linear transformation.

Definition. A function T:\mathbb{R}^n\rightarrow \mathbb{R}^m is called a linear transformation if:
(1) T(\mathbf{x}+\mathbf{y})=T(\mathbf{x})+T(\mathbf{y}); and
(2) T(c\mathbf{x})=cT(\mathbf{x}).
The conditions must be satisfied for all \mathbf{x},\mathbf{y} in \mathbb{R}^n and all c in \mathbb{R}.

Proof.

Let \mathbf{x}=(x_1,x_2) and \mathbf{y}=(y_1,y_2).

Then

\begin{aligned} T(\mathbf{x})& =T(x_1,x_2)=(x_1,x_2,0) \\ T(\mathbf{y})&=T(y_1,y_2)=(y_1,y_2,0) \\ T(\mathbf{x})+T(\mathbf{y}) & = (x_1,x_2,0) + (y_1,y_2,0) \\ \Rightarrow\enspace\textrm{RHS} & = (x_1+y_1,x_2+y_2,0)\\ \mathbf{x}+\mathbf{y} & = (x_1,x_2) + (y_1,y_2) \\ & = (x_1+y_1,x_2+y_2) \\ T(\mathbf{x}+\mathbf{y}) & = T(x_1+y_1,x_2+y_2) \\ \Rightarrow\enspace\textrm{LHS} & = (x_1+y_1,x_2+y_2,0) \\ \therefore\enspace \textrm{LHS} & = \textrm{RHS} \\ \end{aligned}

Condition (1) is thus satisfied.

\forall\, c\in\mathbb{R} and \forall\, \mathbf{x}=(x_1,x_2)\in\mathbb{R}^2,

\begin{aligned} cT(\mathbf{x}) & = cT(x_1,x_2) \\ & = c(x_1,x_2,0) \\ \Rightarrow\enspace\textrm{RHS} & = (cx_1,cx_2,0) \\ T(c\mathbf{x}) & = T(c(x_1,x_2)) \\ & = T(cx_1,cx_2) \\ \Rightarrow \enspace\textrm{LHS} & = (cx_1,cx_2,0) \\ & = \textrm{RHS} \\ \end{aligned}

Condition (2) is also satisfied.

In conclusion, T is a linear transformation.