91Ó°ÊÓ

If matrix \(A\) represents the reflection about a line \(L\) in \(\mathbb{R}^{2},\) what is the dimension of the space \(V\) of all matrices \(S\) such that $$A S=S\left[\begin{array}{rr} 1 & 0 \\ 0 & -1 \end{array}\right] ?$$ Hint: Write \(S=[\vec{v} \vec{w}],\) and show that \(\vec{v}\) must be parallel to \(L\), while \(\vec{w}\) must be perpendicular to \(L\)

A function \(f(t)\) from \(\mathbb{R}\) to \(\mathbb{R}\) is called even if \(f(-t)=\) \(f(t),\) for all \(t\) in \(\mathbb{R},\) and odd if \(f(-t)=-f(t),\) for all \(t .\) Are the even functions a subspace of \(F(\mathbb{R}, \mathbb{R})\), the space of all functions from \(\mathbb{R}\) to \(\mathbb{R} ?\) What about the odd functions? Justify your answers carefully.

Show that the space \(F(\mathbb{R}, \mathbb{R})\) of all functions from \(\mathbb{R}\) to \(\mathbb{R}\) is infinite dimensional.

In this exercise, we will outline a proof of the ranknullity theorem: If \(T\) is a linear transformation from \(V\) to \(W,\) where \(V\) is finite dimensional, then \\[ \begin{aligned} \operatorname{dim}(V) &=\operatorname{dim}(\operatorname{im} T)+\operatorname{dim}(\operatorname{ker} T) \\ &=\operatorname{rank}(T)+\text { nullity }(T) \end{aligned} \\] a. Explain why ker(T) and image ( \(T\) ) are finite dimensional. Hint: Use Exercises 4.1 .54 and 4.1 .57 Now consider a basis \(v_{1}, \ldots, v_{n}\) of \(\operatorname{ker}(T)\) where \(n=\) nullity \((T),\) and a basis \(w_{1}, \dots, w_{r}\) of \(\operatorname{im}(T),\) where \(r=\operatorname{rank}(T) .\) Consider elements \(u_{1}, \ldots, u_{r}\) in \(V\) such that \(T\left(u_{i}\right)=w_{i}\) for \(i=\) \(1, \ldots, r,\) Our goal is to show that the \(r+n\) elements \(u_{1}, \ldots, u_{r}, v_{1}, \ldots, v_{n}\) form a basis of \(V ;\) this will prove our claim. b. Show that the elements \(u_{1}, \ldots, u_{r}, v_{1}, \ldots, v_{n}\) are linearly independent. Hint: Consider a relation \(c_{1} u_{1}+\cdots+c_{r} u_{r}+d_{1} v_{1}+\cdots+d_{n} v_{n}=0,\) ap ply transformation \(T\) to both sides, and take it from there. c. Show that the elements \(u_{1}, \ldots, u_{r}, v_{1}, \ldots, v_{n}\) span \(V .\) Hint: Consider an arbitrary element \(v\) in \(V\), and write \(T(v)=d_{1} w_{1}+\dots+d_{r} w_{r} .\) Now show that the element \(v-d_{1} u_{1}-\dots- d_{r} u_{r}\) is in the kernel of \(T,\) so that \(v-d_{1} u_{1}-\cdots-d_{r} u_{r}\) can be written as a linear combination of \(v_{1}, \ldots, v_{n}\)