Lesson test: Change of basis

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Change of basis

1.[2p]
If the columns of $P$ are the new basis vectors written in standard coordinates, then $P$ converts

new coordinates into standard coordinates

Correct

standard coordinates into new coordinates

The answer is: new coordinates into standard coordinates

vectors into their lengths

The answer is: new coordinates into standard coordinates

any basis into an orthonormal one

The answer is: new coordinates into standard coordinates

2.[2p]
The matrix of the same transformation in the new basis is

$P^{- 1} A P$

Correct

$P A P^{- 1}$

The answer is: $P^{- 1} A P$

$P^{T} A P^{T}$

The answer is: $P^{- 1} A P$

$A P$

The answer is: $P^{- 1} A P$

3.[2p]
With basis $(1, 1)$ and $(- 1, 1)$ , the arrow $(4, 2)$ has coordinates $(a, b)$ . What is $a$ ?

CorrectNot quite: 3

4.[3p]
Which of these are the same for two similar matrices?

Select all that apply

The determinant

Correct

The trace

Correct

The rank

Correct

Whether the matrix is diagonal

The answer is: The determinant, The trace, The rank

5.[3p]
$A = [(2, 1), (1, 2)]$ becomes diagonal in the basis $(1, 1)$ , $(- 1, 1)$ . What is its first diagonal entry there?

CorrectNot quite: 3

6.[2p]
Two matrices with the same trace and the same determinant must be similar.

True

The answer is: False

False

Correct

7.[2p]
The only matrix similar to the identity is the identity itself, because

$P^{- 1} I P = P^{- 1} P = I$ for every invertible $P$

Correct

the identity has determinant one

The answer is: $P^{- 1} I P = P^{- 1} P = I$ for every invertible $P$

the identity is symmetric

The answer is: $P^{- 1} I P = P^{- 1} P = I$ for every invertible $P$

similarity preserves the entries of a matrix

The answer is: $P^{- 1} I P = P^{- 1} P = I$ for every invertible $P$

8.[3p]
Reflecting in the line through $(2, 1)$ has matrix $[(0.6, 0.8), (0.8, k)]$ . What is $k$ ?

CorrectNot quite: -0.6

9.[3p]
Match each requirement to the reason for it.

$P$ must be invertible
An orthonormal basis is preferred
A nearly dependent basis is avoided
The diagonal case is sought

its columns have to form a basis
errors are amplified by the condition number of $P$
it reduces the map to one stretch factor per direction
then the inverse of $P$ is its transpose

Show the answer

$P$ must be invertible: its columns have to form a basis An orthonormal basis is preferred: then the inverse of $P$ is its transpose A nearly dependent basis is avoided: errors are amplified by the condition number of $P$ The diagonal case is sought: it reduces the map to one stretch factor per direction

1.[2p] If the columns of P are the new basis vectors written in standard coordinates, then P converts

2.[2p] The matrix of the same transformation in the new basis is

3.[2p] With basis (1,1) and (−1,1), the arrow (4,2) has coordinates (a,b). What is a?

4.[3p] Which of these are the same for two similar matrices?

5.[3p] A=[(2,1),(1,2)] becomes diagonal in the basis (1,1), (−1,1). What is its first diagonal entry there?

6.[2p] Two matrices with the same trace and the same determinant must be similar.

7.[2p] The only matrix similar to the identity is the identity itself, because

8.[3p] Reflecting in the line through (2,1) has matrix [(0.6,0.8),(0.8,k)]. What is k?

9.[3p] Match each requirement to the reason for it.

1.[2p]
If the columns of $P$ are the new basis vectors written in standard coordinates, then $P$ converts

2.[2p]
The matrix of the same transformation in the new basis is

3.[2p]
With basis $(1, 1)$ and $(- 1, 1)$ , the arrow $(4, 2)$ has coordinates $(a, b)$ . What is $a$ ?

4.[3p]
Which of these are the same for two similar matrices?

5.[3p]
$A = [(2, 1), (1, 2)]$ becomes diagonal in the basis $(1, 1)$ , $(- 1, 1)$ . What is its first diagonal entry there?

6.[2p]
Two matrices with the same trace and the same determinant must be similar.

7.[2p]
The only matrix similar to the identity is the identity itself, because

8.[3p]
Reflecting in the line through $(2, 1)$ has matrix $[(0.6, 0.8), (0.8, k)]$ . What is $k$ ?

9.[3p]
Match each requirement to the reason for it.