Change of Basis

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Change of Basis

Vectors are written as rows in these questions. R^n is the real vector space with n coordinates. Rows of a matrix are separated by semicolons. Enter your answer as concisely as possible.

Let v=(1,2) in the standard basis. What are the coordinates of v in the basis (1,2),(2,3)?

1 point

Let v=(1,0) in the standard basis. What are the coordinates of v in the basis (1,2),(2,3)?

1 point

Let v=(-1,1) in the basis (1,2),(2,3). What are the coordinates of v in the standard basis?

1 point

Let v=(-1,1) and w=(1,-1) in the basis (1,2),(2,3). Is the set {v, w} a basis?

1 point

True

False

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Let A=[1 2 3; 2 3 4; 3 4 5] represent a linear map. If the input and output basis are changed, which of the following will not change in the equivalent matrix?

2 points

rank

nullity

number of nonzero entries in the matrix

Let A=[1 2 3; 2 3 4; 3 4 5] represent a linear map in the standard basis. What is the minimum number of nonzero entries possible in a matrix representing the same linear map w.r.t. to any other input and output basis?

2 points

A linear map represented by an invertible matrix under a change of input and output basis can be represented by the identity matrix.

1 point

True

False

Clear selection

A linear map under a suitable input and output basis can be represented by a diagonal matrix (with zero or nonzero entries on the diagonal).

1 point

True

False

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