True/False: The standard basis for $$ is $$.
True False
True/False: Let $$ be in some vector space $$ and let $$ be a basis for $$. Then $$ can be written in two different ways: where not all of the $$’s are equal to the corresponding $$’s.
True False
Check out the unique representation theorem.

Suppose $$ is a basis for some vector space $$ and $$ is a vector in $$. What is $$?

$$

$$ is a basis for $$ and $$ is a vector in $$. Find the coordinate vector of $$ relative to $$

$$

$$ is a basis for $$ and $$ is a vector in $$. Find the coordinate vector of $$ relative to $$

$$

Let $$ be a basis for $$. If $$, what is $$, that is the same vector in $$, but written in terms of the standard basis of $$?

$$
$$
Let $$ be a basis for $$. If $$ and $$, then which of the following is the basis $$?
$$ $$ $$
$$