91影视

Consider a set $\left\{t^3,t^3+t^2,t^3+t,t^3+1\right\}$.

The set $\left\{\mathbf{1}\mathbf{,}\mathbf{t}\mathbf{,}{\mathbf{t}}_{\mathbf{2}}\mathbf{,}...\mathbf{,}{\mathbf{t}}_{\mathbf{n}}\right\}$ is linear independent set of V if there exist constant ${\mathbf{a}}_{\mathbf{i}}鈭�\mathbf{R}$ such that ${\mathbf{a}}_{\mathbf{0}}\mathbf{+}{\mathbf{a}}_{\mathbf{1}}{\mathbf{t}}_{\mathbf{1}}\mathbf{+}{\mathbf{a}}_{\mathbf{2}}{\mathbf{t}}_{\mathbf{2}}\mathbf{+}...\mathbf{+}{\mathbf{a}}_{\mathbf{n}}{\mathbf{t}}_{\mathbf{n}}\mathbf{=}\mathbf{0}$ where ${\mathbf{a}}_{\mathbf{0}}\mathbf{=}{\mathbf{a}}_{\mathbf{1}}\mathbf{=}{\mathbf{a}}_{\mathbf{2}}\mathbf{=}...\mathbf{=}{\mathbf{a}}_{\mathbf{n}}\mathbf{=}\mathbf{0}$ .

Any polynomial $f\left(t\right)鈭�P_3$ is defined as follows.

$\mathbf{f}\left(\mathbf{t}\right)\mathbf{=}{\mathbf{b}}_{\mathbf{0}}\mathbf{+}{\mathbf{b}}_{\mathbf{1}}\mathbf{t}\mathbf{+}{\mathbf{b}}_{\mathbf{2}}{\mathbf{t}}^{\mathbf{2}}\mathbf{+}...\mathbf{+}{\mathbf{b}}_{\mathbf{n}}{\mathbf{t}}^{\mathbf{n}}$

Consider the constant$b_i$ such that $b_1{t}^3+b_2\left(t^3+t^2\right)+b_3\left(t^3+t\right)+b_4\left(t^3+1\right)=0$.

Simplify the equation $b_1{t}^3+b_2\left(t^3+t^2\right)+b_3\left(t^3+t\right)+b_4\left(t^3+1\right)=0$as follows.

$\begin{array}{rcl}{b}_1{t}^3+b_2\left(t^3+t^2\right)+b_3\left(t^3+t\right)+b_4\left(t^3+1\right)& =& 0\\ \left(b_1+b_2+b_3+b_4\right)t^3+b_2{t}^2+b_{3}t+b_4& =& 0\\ & & \end{array}$

Compare the equation $\left(b_1+b_2+b_3+b_4\right)t^3+b_2{t}^2+b_{3}t+b_4=0$both side as follows.

$$\begin{gathered} \left(b_1+b_2+b_3+b_4\right)t^3+b_2{t}^2+b_{3}t+b_4=\left(0\right)t^3+\left(0\right)t^2+\left(0\right)t+\left(0\right) \\ {b}_1+b_2+b_3+b_4=0 \\ {b}_2=0 \\ {b}_3=0 \\ {b}_4=0 \end{gathered}$$

Substitute the values 0 for $b_2$, for $b_2,$0 , for$b_4$in the equation $b_1+b_2+b_3+b_4=0$as follows.

$$\begin{gathered} b_1+b_2+b_3+b_4=0\backslash \\ {b}_1+\left(0\right)+\left(0\right)+\left(0\right)=0 \\ {b}_1=0 \end{gathered}$$

By the definition of linear independence, the subset $\left\{t^3,t^3+t^2,t^3+t,t^3+1\right\}$is L.I.

Therefore, the set of polynomial $P_3$is spanned by$Span\left\{t^3,t^3+t^2,t^3+t,t^3+1\right\}$ that does not contain any polynomial has degree $猢�2$.

Hence, the statement is false.