Lesson test: Orthogonality and least squares

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Orthogonality and least squares

1.[2p]
Least squares replaces an unsolvable $A 𝐱 = 𝐛$ by

the closest reachable target, found by making the error perpendicular to the column space

Correct

a smaller system obtained by deleting equations

The answer is: the closest reachable target, found by making the error perpendicular to the column space

the system $A 𝐱 = 𝟎$

The answer is: the closest reachable target, found by making the error perpendicular to the column space

an approximation obtained by rounding the data

The answer is: the closest reachable target, found by making the error perpendicular to the column space

2.[2p]
The normal equations are

$A^{T} A \hat{𝐱} = A^{T} 𝐛$

Correct

$A A^{T} \hat{𝐱} = A 𝐛$

The answer is: $A^{T} A \hat{𝐱} = A^{T} 𝐛$

$A \hat{𝐱} = A^{T} 𝐛$

The answer is: $A^{T} A \hat{𝐱} = A^{T} 𝐛$

$A^{T} \hat{𝐱} = 𝐛$

The answer is: $A^{T} A \hat{𝐱} = A^{T} 𝐛$

3.[3p]
Fitting $y = c + m t$ to the readings $y = 1, 3, 4, 6$ at $t = 0, 1, 2, 3$ gives what slope $m$ ?

CorrectNot quite: 1.6

4.[3p]
For that same fit, what is the sum of the squared residuals?

CorrectNot quite: 0.2

5.[2p]
Fitting a horizontal line to the readings $1, 3, 4, 6$ gives $c$ equal to what?

CorrectNot quite: 3.5

6.[2p]
A projection matrix satisfies $P^{2} = P$ .

True

Correct

False

The answer is: True

7.[2p]
If $Q$ has orthonormal columns and is square, then

$Q^{- 1} = Q^{T}$ , and $Q$ preserves every length and angle

Correct

$Q^{- 1} = Q$ , and $Q$ is a projection

The answer is: $Q^{- 1} = Q^{T}$ , and $Q$ preserves every length and angle

$\det Q = 0$

The answer is: $Q^{- 1} = Q^{T}$ , and $Q$ preserves every length and angle

$Q$ must be the identity

The answer is: $Q^{- 1} = Q^{T}$ , and $Q$ preserves every length and angle

8.[2p]
With the orthonormal basis $(0.6, 0.8)$ and $(- 0.8, 0.6)$ , what is the first coordinate of $(2, 3)$ ?

CorrectNot quite: 3.6

9.[3p]
Why do numerical libraries avoid forming $A^{T} A$ ?

It squares the condition number, so a moderately awkward problem becomes unsolvable in double precision

Correct

The product is not symmetric

The answer is: It squares the condition number, so a moderately awkward problem becomes unsolvable in double precision

The product is never invertible

The answer is: It squares the condition number, so a moderately awkward problem becomes unsolvable in double precision

It changes the least squares solution

The answer is: It squares the condition number, so a moderately awkward problem becomes unsolvable in double precision

1.[2p] Least squares replaces an unsolvable A𝐱=𝐛 by

2.[2p] The normal equations are

3.[3p] Fitting y=c+mt to the readings y=1,3,4,6 at t=0,1,2,3 gives what slope m?

4.[3p] For that same fit, what is the sum of the squared residuals?

5.[2p] Fitting a horizontal line to the readings 1,3,4,6 gives c equal to what?

6.[2p] A projection matrix satisfies P2=P.

7.[2p] If Q has orthonormal columns and is square, then

8.[2p] With the orthonormal basis (0.6,0.8) and (−0.8,0.6), what is the first coordinate of (2,3)?

9.[3p] Why do numerical libraries avoid forming ATA?

1.[2p]
Least squares replaces an unsolvable $A 𝐱 = 𝐛$ by

2.[2p]
The normal equations are

3.[3p]
Fitting $y = c + m t$ to the readings $y = 1, 3, 4, 6$ at $t = 0, 1, 2, 3$ gives what slope $m$ ?

4.[3p]
For that same fit, what is the sum of the squared residuals?

5.[2p]
Fitting a horizontal line to the readings $1, 3, 4, 6$ gives $c$ equal to what?

6.[2p]
A projection matrix satisfies $P^{2} = P$ .

7.[2p]
If $Q$ has orthonormal columns and is square, then

8.[2p]
With the orthonormal basis $(0.6, 0.8)$ and $(- 0.8, 0.6)$ , what is the first coordinate of $(2, 3)$ ?

9.[3p]
Why do numerical libraries avoid forming $A^{T} A$ ?