91Ó°ÊÓ

Find the kernel and nullity of the transformation $\mathbf{T}\mathbf{}\mathbf{in}\mathbf{}\mathbf{T}\mathbf{\left(}\mathbf{f}\mathbf{\right)}\mathbf{=}\mathbf{f}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{5}\mathbf{f}\mathbf{\text{'}}\mathbf{+}\mathbf{6}{\mathbf{fC}}^{\mathbf{∞}}$

If the image of a linear transformation T is infinite dimensional, then the domain of T must be infinite dimensional.

Show that a finitely generated space is in fact finite dimensional.

Find the image and kernel of the transformation $\mathbf{T}$in$\mathbf{T}\left(x_0,x_1,x_2,x_3,...\right)\mathbf{=}\left(0,x_0,x_2,x_4\right)$ from $\mathbf{V}$to $\mathbf{V}$.

In this exercise we will show that the functions$\cos \left(x\right)$ and $\sin \left(x\right)$span the solution space$V$ of the differential equation $f\text{'}\text{'}\left(x\right)=-f\left(x\right)$. See Example 1 of this section.

$\left(D-\mathrm{λ}\right)\left(p\left(t\right)e^{\mathrm{λ}t}\right)=p\text{'}\left(t\right)e^{\mathrm{λ}t}$

a. Show that if$g\left(x\right)$ is in$V$ , then the function${\left(g\left(x\right)\right)}^2+{\left(g\text{'}\left(x\right)\right)}^2$ is constant. Hint: Consider the derivative.

b. Show that if $g\left(x\right)$is in$V$ , then the functiondata-custom-editor="chemistry" $g\left(0\right)=g\text{'}\left(0\right)=0$ thendata-custom-editor="chemistry" $g\left(x\right)=0$ for all $x$.

c. If $f\left(x\right)$is in $V$, then $g\left(x\right)=f\left(x\right)-f\left(0\right)\cos x-f\text{'}\left(0\right)\sin x$is in $V$as well (why?). Verify that $g\left(0\right)=0$and $g\text{'}\left(0\right)=0$. We can conclude that $g\left(x\right)=0$for all $\mathit{x}$, so that $f\left(x\right)=f\left(0\right)\cos x+f\text{'}\left(0\right)\sin x$. It follows that the functions $\cos x$and $\sin x$$V$span , as claimed.

If A,B,C and D are noninvertible$\mathbf{2}\mathbf{×}\mathbf{2}$ matrices then the matrices AB,BC, and AD must be linearly dependent.

Show that if 0 is the neutral element of a linear space V then k0=0, for all scalars k.

Find the image, kernel, rank, and nullity of the transformation$\mathbf{T}$ in$\mathbf{T}\mathbf{\left(}\mathbf{f}\mathbf{\right(}\mathbf{t}\mathbf{\left)}\mathbf{\right)}\mathbf{=}\mathbf{f}\mathbf{\left(}\mathbf{7}\mathbf{\right)}$ from ${\mathbf{P}}_{\mathbf{2}}$to $\mathbf{R}$.

Find a basis of a linear space and thus determine its dimension. Examine whether a subset of a linear space is a subspace. Which of the subsets of ${\mathbf{P}}_{\mathbf{2}}$given in Exercises 1through 5are subspaces of${\mathbf{P}}_{\mathbf{2}}$ (see Example 16)? Find a basis for those that are subspaces,$\mathbf{\{}\mathbf{p}\left(t\right)\mathbf{:}\mathbf{p}\left(-t\right)\mathbf{=}\mathbf{-}\mathbf{p}\left(t\right)\mathbf{,}\mathbf{}\mathbf{for}\mathbf{}\mathbf{all}\mathbf{}\mathbf{t}\mathbf{\}}$.

Find the transformation is linear and determine whether they are isomorphism.