Homework 4

(ADDITIONAL PROBLEMS)

Problem 1   Let $T:V\to W$ be a linear transformation.

a) Let $U\subseteq V$ be a subspace of $V$. Show that $T(U)\doteq \{w\in W\, :\, w=T(u)\, \textrm{for some }\, \, u\in U\}$ is a subspace of $W$. If $U=V$, then $T(V)$ is the subspace called the range of the linear transformation $T$.

b) Let $X\subseteq W$ be a subspace of $W$. Show that $T^{-1}(X)\doteq \{v\in V\, :\, T(v)\in X\}$ is a subspace of $V$. If $X=\{0\}$, then $T^{-1}(\{0\})$ is the subspace called the null space of $T$.

Problem 2   Let $T:\mathbb{R}^{2}\to \mathbb{R}^{2}$ be the reflection through the line $y=\alpha x$ on the plane. Find the matrix of this transformation with respect to the standard basis.

Problem 3   Recall the following facts:

• Two non-zero vectors in $\mathbb{R}^{3}$ are perpendicular iff their dot product is $0$;
• The cross product of two linearly independent vectors in $\mathbb{R}^{3}$ is perpendicular to the plane spanned by those vectors in $\mathbb{R}^{3}$.

Using these facts, solve the following problem: Let $v=(1,1,1)$, $w=(3,2,1)$ and $x=(-1,0,1)$ be vectors in $\mathbb{R}^{3}$. Let $T:\mathbb{R}^{3}\to \mathbb{R}^{3}$ be the linear transformation defined as the composition of the following:
1) first, perform rotation in the plane perpendicular to $v$ by angle $\theta$;
2) then, do a flip in the plane perpendicular to $w$;
3) finally, do a (counterclockwise) rotation in the plane perpendicular to $x$ by angle $\varphi$;
Determine the matrix of this linear transformation in the standard basis in $\mathbb{R}^{3}$. (Do not multiply the three matrices you will have).

Problem 4   Let $D:P_{n}(\mathbb{R})\to P_{n-1}(\mathbb{R})$ be the differentiation transformation defined by $(Df)(x)=f'(x)$ and $I:P_{n}(\mathbb{R})\to P_{n+1}(\mathbb{R})$ be the integration transformation defined by $(If)(x)=\int _{0}^{x}f(t)dt$. Find the matrices of these transformations with respect to the standard bases in $P_{n-1}(\mathbb{R}),\, P_{n}(\mathbb{R}),\, P_{n+1}(\mathbb{R})$.

Problem 5   Let $V,U,W$ be vector spaces such that $V=U\oplus W$. Let $T:V\to V$ be a linear transformation such that $U$ is invariant under $T$. Is it true that $W$ is also invariant under $T$? Prove or give a counterexample.

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Homework 4

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