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Section 2.2 Linear Mappings

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Checkpoint 2.2.1.

For each of the following functions, determine whether or not it is linear. If a given function is linear, show that it satisfies properties

(2.2.1) and

(2.2.2). If a given function is not linear, give a specific example to show that it fails at least one of the properties

(2.2.1) or

(2.2.2).

the function that sends every input point to its reflection across the \(x\)-axis

the function \(g\) given by \(g(a,b)=(a+b,\sqrt{a^2+b^2})\)

the constant function that sends every input point to the origin

the constant function that sends every input point to the point \((1,0)\)

the function \(h\) given by \(h(a,b)=(a+b,a-b)\)

the function that sends input vector \(\mathbf{x}\) to output \(\mathbf{x}+\mathbf{b}\text{,}\) where \(\mathbf{b}\) is a constant nonzero vector

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Checkpoint 2.2.2.

Suppose that \(f\colon \R^2\to\R^2\) is given by \(f(x,y)=(ax+by,cx+dy)\) for some real constants \(a,b,c,d\text{.}\) Show that \(f\) is linear.

Suppose that the matrix for a linear map \(L\) is \(\twotwo{2}{1}{1}{-1}\text{.}\) Find \(L(3,2)\text{.}\)

Suppose that a linear map \(L\) satisfies \(L(2,0)=(3,4)\) and \(L(0,3)=(2,1)\text{.}\) Find the matrix for \(L\text{.}\)

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Checkpoint 2.2.3.

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Checkpoint 2.2.4.

Verify that

\(\alpha L\text{,}\) \(L+M\text{,}\) and

\(LM\) are indeed linear maps, that is, that they satisfy properties

(2.2.1) and

(2.2.2).

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Checkpoint 2.2.5.

Choose random integer values between

\(-5\) and

\(+5\) for the constants

\(\alpha,a,b,c,d,e,f,g,h,x,y\text{.}\) Set

\(S=\twotwo{a}{b}{c}{d}\text{,}\) let

\(T=\twotwo{e}{f}{g}{h}\text{,}\) and let

\(\mathbf{u}=\begin{bmatrix}x\\y\end{bmatrix}\text{.}\) Calculate

\(\alpha
S\text{,}\) \(S+T\text{,}\) \(ST\text{,}\) \(TS\text{,}\) \(S\mathbf{u}\text{,}\) and

\(T\mathbf{u}\text{.}\) Check your answers with an online matrix algebra calculator. Repeat the exercise until you can do matrix multiplication quickly and accurately, without having to look at the formula

(2.2.16).

Find a set of values for \(a,b,c,d,e,f,g,h\) for which \(ST\neq TS\text{.}\) Find another set of values for \(a,b,c,d,e,f,g,h\text{,}\) all of them nonzero, for which \(ST=TS\text{.}\)

*Comment:* The quantity \(ad-bc\) is called the determinant of the matrix \(\twotwo{a}{b}{c}{d}\text{,}\) denoted \(\det\left(\twotwo{a}{b}{c}{d}\right)\).

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Checkpoint 2.2.6.

Verify that the identity function \(\Id\colon \R^2\to \R^2\) is linear, and that the matrix for the identity function is \(\twotwo{1}{0}{0}{1}\text{.}\)

Verify each of the equalities in

(2.2.17).

Explain each of the connecting phrases "which is the same as", "it follows that", and "we conclude that" immediately following equations

(2.2.19),

(2.2.20), and

(2.2.21).

## Hint.