Section 1.8

1. Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L,K:V\to V} be linear maps between finite-dimensional vector spaces that satisfy Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L\circ K=0} . Is it true that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle K\circ L=0} ?

Solution:

No. in general composition of functions is not commutative. By the theorem that any linear map can be expressed as a matrix, finding a counterexample comes down to finding two matrices Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A,B} such that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle AB=0} but Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle BA\neq 0} . Here is one example of functions: $V=\mathbb {R} ^{2}$. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L(x,y)=(x,0),K(x,y)=(0,x+y)} . Then we have Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L\circ K(x,y)=L(0,x+y)=(0,0)} but Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle K\circ L(x,y)=K(x,0)=(0,x+0)} so Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L\circ K=0} but Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle K\circ L\neq 0} .

4. Show that a linear map Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L:V\to W} is one-to-one if and only if Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L(x)=0} implies $x=0$.

Proof:

First note that for any linear map Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L(0)=0} because $L(0)=L(0\cdot 0)=0\cdot L(0)=0$. Proof: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (\Rightarrow )} Suppose that $L$ is one-to-one. Then if Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L(x)=0} we have Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L(x)=L(0)} by the note above so that we must have $x=0$. Therefore Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L(x)=0} implies $x=0$. Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle (\Leftarrow )} Now suppose that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L(x)=0} implies $x=0$. If Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L(x)=L(y)} then by linearity of $L$ we have Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L(x-y)=L(x)-L(y)=0} . But then by hypothesis that means Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x-y=0} which implies Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=y} . Therefore $L$ is one-to-one.

6. Let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle V\neq \{0\}} be finite-dimensional and assume that

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L_{1},L_{2},...,L_{n}:V\to V}

are linear operators. Show that if Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L_{1}\circ L_{2}\circ \cdots \circ L_{n}=0} then at least one of the Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L_{i}} are not one-to-one.

Proof:

I will use proof by contrapositive. The equivalent statement would then be "`If all of the Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L_{i}} are one-to-one, then Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L_{1}\circ \cdots \circ L_{n}\neq 0} . Then this becomes very easy if you know the fact from set theory that the composition of one-to-one functions is a one-to-one function. This gives the following. Suppose that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L_{1},...,L_{n}} are all one-to-one. Then $L_{1}\circ \cdots \circ L_{n}$ is also a one-to-one function and so the only input that will give an output of 0 is the input $0$ from problem 4. Therefore Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle L_{1}\circ \cdots \circ L_{n}\neq 0} and we are done. If you don’t know the fact from set theory you can prove it as follows. Suppose $f,g$ are one-to-one functions. Consider the function $f\circ g$. Then to show this new function is one-to-one assume that Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f\circ g(x)=f\circ g(y)} . Then Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle f(g(x))=f(g(y))} . But since $f$ is one-to-one that means the inputs to $f$ must be the same or in other words Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g(x)=g(y)} . But then $g$ is one-to-one so that means Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle x=y} and therefore $f\circ g$ is one-to-one.

13. Consider the map Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Psi :\mathbb {C} \to {\text{Mat}}_{2\times 2}(\mathbb {R} )} defined by Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Psi (\alpha +i\beta )={\begin{bmatrix}\alpha &-\beta \\\beta &\alpha \end{bmatrix}}} ( a) Show that this is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {R} } -linear and one-to-one, but not onto. Find an example of a matrix in ${\text{Mat}}_{2\times 2}(\mathbb {R} )$ that does not come from $\mathbb {C}$.

Proof:

To show this is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {R} -} linear let Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle z_{1}=\alpha _{1}+i\beta _{1},z_{2}=\alpha _{2}+i\beta _{2}\in \mathbb {C} } . Then: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Psi (z_{1}+z_{2})=\Psi (\alpha _{1}+i\beta _{1}+\alpha _{2}+i\beta _{2})=\Psi (\alpha _{1}+\alpha _{2}+i(\beta _{1}+\beta _{2}))={\begin{bmatrix}\alpha _{1}+\alpha _{2}&-\beta _{1}-\beta _{2}\\\beta _{1}+\beta _{2}&\alpha _{1}+\alpha _{2}\end{bmatrix}}={\begin{bmatrix}\alpha _{1}&-\beta _{1}\\\beta _{1}&\alpha _{1}\end{bmatrix}}+{\begin{bmatrix}\alpha _{2}&-\beta _{2}\\\beta _{2}&\alpha _{2}\end{bmatrix}}=\Psi (z_{1})+\Psi (z_{2})} Similarly if $z=\alpha +i\beta \in \mathbb {C}$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle a\in \mathbb {R} } then: Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Psi (az)=\Psi (a\alpha +ia\beta )={\begin{bmatrix}a\alpha &-a\beta \\a\beta &a\alpha \end{bmatrix}}=a{\begin{bmatrix}\alpha &-\beta \\\beta &\alpha \end{bmatrix}}=a\Psi (z)} Therefore Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Psi } is Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \mathbb {R} } -linear. Now to show Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \Psi } is not onto we notice that any matrix in the image of $\Psi$ has top left and bottom right coordinate the same. So the simple matrix Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{bmatrix}1&2\\3&4\end{bmatrix}}} cannot possibly be in the image of $\Psi$. Therefore $\Psi$ is not onto.