Outline of Student Process in Question 2
In this section, you will be debugging the following Gram Schmidt Orthonormalization and QR Decomposition process
made by a student.
Problem: Solve for its QR Decomposition of
A =


3 −3 1
4 −4 7
0 3 3



Solution: Finding the first column of Q:

⃗q1 =


3
4
0



Finding the second column of Q:

proj ⃗q1
( ⃗a2) = ( ⃗a2 · ⃗q1)⃗q1
⃗z2 = ⃗a2 − proj ⃗q1
( ⃗a2)

=


−3
−4
3

 −


−3
−4
0

 =


0
0
3



⃗q2 =


0
0
3



Finding the third column of Q:

proj{ ⃗q1, ⃗q2}
( ⃗a3) = ( ⃗a3 · ⃗q1)⃗q1 + ( ⃗a3 · ⃗q2)⃗q2
=


−3
−4
3


⃗z3 = ⃗a3 − proj{ ⃗q1, ⃗q2}
( ⃗a3)

=


1
−7
3

 −


−3
−4
3

 =


4
−3
0



⃗q3 =


4
−3
0


And now, finding out that we still have to find the matrix R:

R =


⃗a1 · ⃗q1 ⃗a1 · ⃗q2 ⃗a1 · ⃗q3
0 ⃗a2 · ⃗q2 ⃗a2 · ⃗q3
0 0 ⃗a3 · ⃗q3

 =


25 0 0
0 9 0
0 0 −17

 whats wrong w this

The student's solution to finding the QR decomposition of the matrix \( A \) has several errors in both the Gram-Schmidt Orthonormalization process and the computation of the matrix \( R \). I'll go through the mistakes step by step.

### Matrix \( A \):
The matrix \( A \) is:
\[
A =
\begin{bmatrix}
3 & -3 & 1 \\
4 & -4 & 7 \\
0 & 3 & 3
\end{bmatrix}
\]

### Step 1: Finding the first column of \( Q \) (\(\vec{q_1}\))

The student correctly identifies \( \vec{q_1} \) as:
\[
\vec{q_1} =
\begin{bmatrix}
3 \\
4 \\
0
\end{bmatrix}
\]

However, \( \vec{q_1} \) should be normalized to obtain an orthonormal vector. The normalization is done as follows:

\[
\| \vec{q_1} \| = \sqrt{3^2 + 4^2 + 0^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]

Thus, the correct normalized \( \vec{q_1} \) is:

\[
\vec{q_1} = \frac{1}{5}
\begin{bmatrix}
3 \\
4 \\
0
\end{bmatrix}
=
\begin{bmatrix}
\frac{3}{5} \\
\frac{4}{5} \\
0
\end{bmatrix}
\]

### Step 2: Finding the second column of \( Q \) (\(\vec{q_2}\))

The student attempts to find \( \vec{q_2} \) by projecting \( \vec{a_2} \) onto \( \vec{q_1} \):

\[
\text{proj}_{\vec{q_1}}(\vec{a_2}) = \frac{\vec{a_2} \cdot \vec{q_1}}{\vec{q_1} \cdot \vec{q_1

Explore the debugging process of Gram-Schmidt Orthonormalization and QR Decomposition applied to the matrix A = [3, -3, 1; 4, -4, 7; 0, 3, 3]. Identify errors in steps including vector projection and matrix R computation. Suitable for undergraduate level students studying linear algebra.

Math Problem Statement

proj{ ⃗q1, ⃗q2} ( ⃗a3) = ( ⃗a3 · ⃗q1)⃗q1 + ( ⃗a3 · ⃗q2)⃗q2

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