Exercises

For each of the following functions, determine whether the axioms 1 and 2 are satisfied.

1. i.

   $T:{ℝ}^3\to ℝ$, where $T(x,y,z):=x+y+1$.

2. ii.

   $D:{𝒫}_3(ℝ)\to {𝒫}_3(ℝ)$, where $D(f):=\frac{df}{dx}$; in other words, $D$ is defined by differentiating real polynomials which are degree less than or equal to 3.

3. iii.

   $\mathrm{tr}:{\mathrm{M}}_3(ℂ)\to ℂ$ defined by $\mathrm{tr}(A):=a_{11}+a_{22}+a_{33}$; this is the trace of a matrix, defined by adding together the entries on the diagonal.

Let $T:{ℝ}^2\to {ℝ}^2$ be defined by $T(x,y)=(2x-y,x+3y)$. Let $𝒞$ be the standard basis, and $ℬ=((1,0),(1,1))$.

1. i.

   Compute ${}_{𝒞}[T]_{𝒞}$, ${}_{ℬ}[T]_{𝒞}$, ${}_{𝒞}[T]_{ℬ}$, and ${}_{ℬ}[T]_{ℬ}$,

2. ii.

   Compute ${}_{ℬ}[T\circ T]_{ℬ}$,

3. iii.

   Hence verify that $({}_{ℬ}[T]_{𝒞})({}_{𝒞}[T]_{ℬ})={}_{ℬ}[T\circ T]_{ℬ}=({}_{ℬ}[T]_{ℬ})({}_{ℬ}[T]_{ℬ})$.

Let $T:{\mathrm{M}}_2(ℝ)\to {ℝ}^2$ be . If $𝒞$ is the standard basis of ${ℝ}^2$ and $ℬ$ is the standard basis of ${\mathrm{M}}_2(ℝ)$:

Compute ${}_{𝒞}[T]_{ℬ}$.

For each of the following linear transformations, find a basis of the image and for the kernel. Hence verify the result of Theorem 4.24 in these cases.

1. i.

   .

2. ii.

   $T:{ℝ}^2\to {ℝ}^2$ defined by $T(x,y)=(x+y,x+y)$.

3. iii.

   $T:{ℂ}^2\to {ℂ}^3$ defined by $T(x,y)=(x+2iy,y-x,ix+y)$.

4. iv.

   $T:{ℝ}^3\to {𝒫}_2(ℝ)$ defined by $T(a,b,c)=(a-b)+(b-c)x+(a-c)x^2$.

Let $T(x,y,z)=(2x-y-z,2y-x-z,2z-x-y)$ be a linear transformation from ${ℝ}^3\to {ℝ}^3$, and let $𝒞$ be the standard basis. So . For each of the following bases of ${ℝ}^3$, find ${}_{𝒞}[\mathrm{Id}]_{ℬ}$, and then use Theorem 4.52 to find the matrix ${}_{ℬ}[T]_{ℬ}$.

1. i.

   $ℬ=((1,1,0),(1,0,1),(0,1,1)),$

2. ii.

   $ℬ=((1,1,0),(1,2,0),(1,2,1)),$

3. iii.

   $ℬ=((1,1,1),(2,3,2),(1,5,4)).$

Let $T:V\to V$ be a linear transformation, and assume $\lambda$ is an eigenvalue of $T$.

1. i.

   Prove that if $T^r$ is the identity transformation then ${\lambda }^r=1$.

2. ii.

   Prove that if $T^2=T$ (in other words, $T$ is idempotent) then $\lambda =0$ or 1.

3. iii.

   Prove that if $T^r=0$ (in other words, $T$ is nilpotent) then $\lambda =0$.

Prove that similarity defines an equivalence relation on ${\mathrm{M}}_n(F)$. In other words, for $A,B,C\in {\mathrm{M}}_n(F)$:

1. i.

   (Reflexivity): Prove that $A$ is similar to $A$.

2. ii.

   (Symmetry): Prove that if $A$ is similar to $B$, then $B$ is similar to $A$.

3. iii.

   (Transitivity): Prove that if $A$ is similar to $B$, and $B$ is similar to $C$, then $A$ is similar to $C$.

Let $A,B\in {\mathrm{M}}_n(F)$.

1. i.

   Prove that $A$ is invertible if and only if $\mathrm{rank}A=n$.

2. ii.

   Prove that if $A$ is invertible then $\mathrm{rank}AB=\mathrm{rank}B$

Let $A,B\in {\mathrm{M}}_n(F)$, prove that $\mathrm{rank}AB\le \min \{\mathrm{rank}A,\mathrm{rank}B\}$.

Find examples of the following (possibly non-linear) functions:

1. i.

   $f:{ℝ}^3\to {ℝ}^2$ such that $S:=\{(x,y,z)\in {ℝ}^3|f(x,y,z)=(0,0)\}$ is a subspace.

2. ii.

   $f:{ℝ}^2\to {ℝ}^3$ such that $S:=\{(x,y)\in {ℝ}^2|f(x,y)=(0,0,0)\}$ is a 2-dimensional subspace.

3. iii.

   $f:{ℝ}^2\to {ℝ}^2$ such that $S:=\{(x,y)\in {ℝ}^2|f(x,y)=(0,0)\}$ is the empty set.

4. iv.

   $f:{ℝ}^2\to ℝ$ such that $S:=\{(x,y)\in {ℝ}^2|f(x,y)=0\}$ is not the empty set, and is also not a subspace.

Assume that $T,S:V\to V$ are bijective linear transformations between vector spaces (possibly infinite dimensional). Prove $T\circ S:V\to V$ is a bijective linear transformation with ${(T\circ S)}^{-1}=S^{-1}\circ T^{-1}$.

[Recall, $\circ$ means “compose” the transformations.]

Recall that $V=ℂ$ may be viewed as a 2-dimensional vector space over the field $ℝ$, and we can use the standard basis $ℬ=\{1,i\}$. The function $T:ℂ\to ℂ$ which sends $x\mapsto ix$ is a linear transformation of the 2-dimensional real vector space $ℂ$.

1. i.

   Find the $2\times 2$ matrix $A={[T]}_{ℬ}$.

2. ii.

   Prove that $T$ is a linear transformation of 1-dimensional complex vector spaces.

3. iii.

   Can you find a $2\times 2$ matrix which produces a real linear transformation $ℂ\to ℂ$, which is not a complex linear transformation $ℂ\to ℂ$?

A linear transformation $T:V\to V$ on an inner product space is called self-adjoint if:

for all $\overrightarrow{x},\overrightarrow{y}\in V$.

1. i.

   Prove that $T:{ℝ}^n\to {ℝ}^n$ (using the standard inner product) is self-adjoint if and only if its associated matrix (in the standard basis) is symmetric.

2. ii.

   Let $V$ be the inner product space of real-valued continuous functions on $[0,1]$ from Example 3.15. Consider the function $g(t)=t$, which is in $V$. Then define $T:V\to V$ by $T(f):=g\cdot f$. Prove that $T$ is self-adjoint.

3. iii.

   Prove that $T$ from part (ii) has no eigenvalues nor eigenvectors.

This example proves that the spectral decomposition from Theorem 5.7 does not generalize to self-adjoint linear transformations of infinite-dimensional inner product spaces.

Let $V$ be the vector space of all functions $ℝ\to ℝ$ which are infinitely differentiable everywhere (also called $C^{\mathrm{\infty }}$, meaning, their $n^{th}$-derivatives exist for any $n$). Then differentiation defines a map $D:V\to V$.

1. i.

   Verify that $D$ is a linear transformation

2. ii.

   What is the kernel of $D$?

3. iii.

   What is the image of $D$?

(Bonus of Pisa) Let , and $\overrightarrow{x_1}=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \end{array}\right]$. Inductively define a sequence of vectors $\overrightarrow{x_i}=A\overrightarrow{x_{i-1}}$, for all $i\ge 2$.

1. i.

   Write down the vectors $\overrightarrow{x_1},\cdot ,\overrightarrow{x_8}$.Do you see a pattern?

2. ii.

   Find a diagonal matrix $D$ and invertible matrix $P$ such that $A=PD{P}^{-1}$.

3. iii.

   Use the equation $\overrightarrow{x_n}=A^{n-1}\overrightarrow{x_1}=(P^{-1}{D}^{n-1}P)\overrightarrow{x_1}$ to devise an explicit formula for the coordinates of $\overrightarrow{x_n}$.