We are tasked with proving that the linear map $[ \cdot ]_\alpha : V \to \mathbb{R}^n$ defined by

$[x]_\alpha = (a_1, a_2, \ldots, a_n) \quad \text{where} \quad x = a_1 x_1 + a_2 x_2 + \cdots + a_n x_n$

is bijective. This involves showing that the map is both injective and surjective.

1. Injectivity

To prove that the map is injective, we must show that if $[x]_\alpha = [y]_\alpha$, then $x = y$.

• Let $x, y \in V$ be two elements such that $[x]_\alpha = [y]_\alpha$. This means that the coordinate vectors of $x$ and $y$ with respect to the basis $\alpha = \{x_1, x_2, \dots, x_n\}$ are the same. That is,

[ [x]\alpha = (a_1, a_2, \ldots, a_n) \quad \text{and} \quad [y]\alpha = (b_1, b_2, \ldots, b_n), ]

where

$x = a_1 x_1 + a_2 x_2 + \cdots + a_n x_n \quad \text{and} \quad y = b_1 x_1 + b_2 x_2 + \cdots + b_n x_n.$

• Since $[x]_\alpha = [y]_\alpha$, it follows that

$(a_1, a_2, \ldots, a_n) = (b_1, b_2, \ldots, b_n),$

so $a_i = b_i$ for all $i = 1, 2, \ldots, n$. This implies that

$x = a_1 x_1 + a_2 x_2 + \cdots + a_n x_n = b_1 x_1 + b_2 x_2 + \cdots + b_n x_n = y.$

Thus, $x = y$. Hence, the map is injective.

2. Surjectivity

To prove that the map is surjective, we need to show that for every vector $v = (v_1, v_2, \ldots, v_n) \in \mathbb{R}^n$, there exists an element $x \in V$ such that $[x]_\alpha = v$.

• Let $v = (v_1, v_2, \ldots, v_n)$ be an arbitrary vector in $\mathbb{R}^n$.
• Consider the vector $x \in V$ defined by

$x = v_1 x_1 + v_2 x_2 + \cdots + v_n x_n.$

• The coordinates of $x$ with respect to the basis $\alpha = \{x_1, x_2, \dots, x_n\}$ are precisely $(v_1, v_2, \ldots, v_n)$, i.e.,

$[x]_\alpha = (v_1, v_2, \ldots, v_n).$

• Therefore, for every vector $v \in \mathbb{R}^n$, there exists an element $x \in V$ such that $[x]_\alpha = v$. Hence, the map is surjective.

Conclusion

Since the map $[ \cdot ]_\alpha$ is both injective and surjective, it is bijective.

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Would you like further clarifications or details on any step? Here are some related questions:

1. What is the general definition of a bijective linear map?
2. How does the dimension of a vector space affect injectivity and surjectivity of a map?
3. What is the significance of choosing a basis for a vector space in defining coordinate maps?
4. How can we extend this argument to infinite-dimensional vector spaces?
5. What is the relationship between linear independence and injectivity?

Tip: When proving injectivity, always check if the kernel of the map only contains the zero vector, as this is equivalent to injectivity.