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QBUS1040 Linear independence
Creation date：2024-06-03 14:39:04
Belonging category：数学mathematics
Linear independence

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QBUS1040

Linear independence
Question 5
Consider the following diagram:
q1
q˜2
a
−βq1
It is known that ∥q1∥ = 1.
(a) Write q˜2 in terms of β, a and q1.
q˜2 = a− βq1
(b) Given that q1 ⊥ q˜2 i.e. qT1 q˜2 = 0, derive an expression for β in terms of a and q1.
0 = qT1 q˜2 = q
T
1 (a− βq1) = qT1 a− β qT1 q1︸︷︷︸
1
= qT1 a− β → β = qT1 a
Question 6
In the Gram-Schmidt algorithm we make the calculation:
q˜i = ai − (qT1 ai)q1 − · · · − (qTi−1ai)qi−1
What does this equation look like for:
(a) q˜1? q˜1 = a1
(b) q˜2? q˜1 = a2 − (qT1 a2)q1
(c) q˜3? q˜3 = a3 − (qT1 a3)q1 − (qT2 a3)q2
Question 7
Given that qi =
q˜i
∥q˜i∥ → qi∥q˜i∥ = q˜i, we can rewrite the equation from the previous question like so:
qi∥q˜i∥ = ai − (qT1 ai)q1 − · · · − (qTi−1ai)qi−1
(a) Rearrange the equation to make ai the subject.
ai = qi∥q˜i∥+ (qT1 ai)q1 + · · ·+ (qTi−1ai)qi−1
(b) Looking at the equation from part (a), circle the terms which are scalars.
ai = qi∥q˜i∥+ (qT1 ai)q1 + · · ·+ (qTi−1ai)qi−1
(c) Re-write the equation from part (a) replacing every scalar with a βi term.
ai = βiqi + β1q1 + · · ·+ βi−1qi−1
(d) Is ai a linear combination of q1, . . . qi? Yes.
Question 8
Similar to the re-arranging in the previous question, we can also get the formula:
qi = (1/∥q˜i∥)(ai − (qT1 ai)q1 − · · · − (qTi−1ai)qi−1)
Using this formula:
(a) What is q1? q1 = (1/∥q˜1∥)︸ ︷︷ ︸
scalar
a1
(b) What is q2 as a linear combination of a1 and a2? (Hint: substitute in your answer from part a)
q2 = (1/∥q˜2∥)
(
a2 − (qT1 a2)q1
)
= (1/∥q˜2∥)︸ ︷︷ ︸
scalar
a2 − (qT1 a2)(1/∥q˜1∥)(1/∥q˜2∥)︸ ︷︷ ︸
scalar
a1
Page 1 of 2.
QBUS1040 Lecture 5: Question Sheet Solutions Semester 2, 2023
Matrices and matrix examples
Question 1
Consider the matrix A =
[
2 6 8
3 −1 0
]
:
(a) What is the size of the matrix A? 2× 3
(b) What is A1,2? 6
(c) Is A a tall, wide or square matrix? wide
(d) What is the matrix given by A1:2,2:3?
[
6 8
−1 0
]
Question 2
Spot the difference! Below we have written the vector x as both a row vector and a column vector.
Circle all of the differences in the notation that you can see.
x = (2,− 4,3,8) x = [2 − 4 3 8]
Question 3
(a) Given the vectors a1 = (2, 3), a2 = (1, 0) and a2 = (4, 4), what does the matrix A = [a1 a2 a3]
look like?
a1, a2 and a3 are column vectors. This means that A =
[
2 1 4
3 0 4
]
.
(b) Given the vectors a1 = [2 3], a2 = [1 0] and a2 = [4 4], what does the matrix A = [a1 a2 a3]
look like?
a1, a2 and a3 are row vectors. This means that A =
[
2 3 1 0 4 4
]
.
Question 4
(a) What does the matrix 03×2 look like?
0 00 0
0 0

(b) What does the matrix I2 look like?
[
1 0
0 1
]
(c) What does the matrix diag(3,−2) look like?
[
3 0
0 −2
]
Page 2 of 2.
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SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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