Recall that the standard bases for $ \mathbb{R}^2 $ and $ \mathbb{R}^3 $ are {(1,0), (0,1)} and {(1,0,0),(0,1,0),(0,0,1)} respectively. In parallel fashion, standard bases for $P_1$ and $P_2$ are {1, x} and {1, x, $x^2$}, respectively.
a)Find a different basis for each of the four vector spaces: $ \mathbb{R}^2, \mathbb{R}^3, P_1$, and $ P_2 $.
b)Is {$\frac{-1}{2}, 8.7x$} a basis for $P_1$?
c) Is {$(3, -4), (-7,5)$} a basis for $ \mathbb{R}^2$?
The answers, according to the professor, are:
a) Examples: {(2, 3), (-7, 8)} for $ \mathbb{R}^2$ and for $P_1$ {$2x +3, -7x + 8$}
{$(3,2,4), (-2,0,8), (-1,4,5)$} for $ \mathbb{R}^3$ and for $P_2$ {$3x^2 + 2x+4, -2x^2+8, -x^2 + 4x+5$}
b) Yes, 2-lin. indep. vectors in a 2-dimensional space.
c) Yes. 2-lin. indep. vectors in a 2-dimensional space.
I have no idea what this problem is asking and how to solve it. Any help would be greatly appreciated. Note: I don't know how to make the correct R in MathJax, the R that stands for all real numbers. I have been stuck on this for a while now, please help.
Note that the dimension of $\mathbb{R^2}$ is $2$ and the dimension of $\mathbb{R^3}$ is $3$. As you have mentioned, the standard bases are $\begin{bmatrix} 0\\ 1 \end{bmatrix}$$\begin{bmatrix} 1\\ 0 \end{bmatrix}$ for $\mathbb{R^2}$. So, we only need two vectors to span the entire vector space.
Furthermore, this implies that we only need any two linearly independent vectors to span $\mathbb{R^2}$ and three linearly independent vectors for $\mathbb{R^3}$.
A vector is linearly independent from other vectors in the set if it cannot be written as a linear combination of the other vectors in the set.
For example, a linear combination of $\begin{bmatrix} 0\\ 1 \end{bmatrix}$ would be $\begin{bmatrix} 0\\ c \end{bmatrix}:c\in \mathbb{R}$
Now, the vector spaces $P_1$ and $P_2$ have dimensions $2$ and $3$, respectively. This is easy to relate to the vector spaces $\mathbb{R^2}$ and $\mathbb{R^3}$.
So, any vector space of dimension $2$ can be spanned by two linearly independent vectors and $P_1$ can be generically represented as $a+bx:a,b\in \mathbb{R}$. A possible set of vectors that span $P_1$ is $\{1,0\cdot x\},\{0,x\}$