True/False: The standard basis for is .

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.

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 ?