Understanding the solution to this subspace problem in linear algebra

[MaxJ]

Homework Statement

Below

Relevant Equations

Below

For this problem,

The solution for (a) is

I am slightly confused for $p \in W$ since I get $a_3 = 2a_1$ and $a_2 = 2a_0$. Since $a_3 = 2b$, $a_2 = 2a$, $a_1 = b$, $a_0 = a$.

Anybody have this doubt too?

Kind wishes

 

[Mark44]

I agree with your conclusion and assume that whoever wrote what you attached made a typo. Further evidence is that the author also wrote "Thus p(z) ..." when clearly p(x) was intended.

 

[fresh_42]

Homework Statement: Below
Relevant Equations: Below

For this problem,
View attachment 349805
The solution for (a) is
View attachment 349806
I am slightly confused for $p \in W$ since I get $a_3 = 2a_1$ and $a_2 = 2a_0$. Since $a_3 = 2b$, $a_2 = 2a$, $a_1 = b$, $a_0 = a$.

Anybody have this doubt too?

Kind wishes

I got the same result as claimed:
\begin{align*}
a+ax+bx^2+bx^3&=\lambda c+\lambda dx+2\lambda cx^2+2\lambda dx^3\\
c&=\lambda^{-1}a\\
d&=\lambda^{-1}a=c\\
a+ax+bx^2+bx^3&=a+ax+2ax^2+2ax^3\\
b&=2a \\[6pt]
U\cap W&=\{a(1+x+2x^2+2x^3)\,|\,a\in \mathbb{F})\}
\end{align*}
but I get confused with the $a_i$ in the book.