PyPDF2 import PdfReader

Load the uploaded PDF and extract text from all pages

pdf_path = '/mnt/data/1738417645242y0zxj7at.pdf' reader = PdfReader(pdf_path)

Extract text from all pages

pdf_text = '' for page in reader.pages: pdf_text += page.extract_text()

Display the first 2000 characters to check the content

pdf_text[:2000]It looks like you've uploaded a document related to a math tutorial involving vector spaces, linear independence, and subspaces. Based on the content, question 28 doesn't appear to be listed directly in the first few lines I retrieved.

I will continue searching through the document for question 28 specifically. Searching for "question 28" in the extracted text to locate the relevant question. import re

Search for occurrences of "28" to identify the exact location of question 28

question_28_start = re.search(r"28", pdf_text)

Extracting content starting from question 28

if question_28_start: question_28_text = pdf_text[question_28_start.start():question_28_start.start() + 2000] else: question_28_text = "Question 28 not found in this range."

question_28_textQuestion 28 in your document is about finding the rank, row space, and column space of matrices. Here is the problem description:

28. Find the rank. Find a basis for the row space. Find a basis for the column space.

(a)

2 & 4 & 6 \\ 4 & 0 & 2 \\ 0 & 2 & 6 \\ \end{pmatrix}$$ (b) $$\begin{pmatrix} 2 & 6 & 6 \\ 4 & 9 & 0 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \\ \end{pmatrix}$$ (c) $$\begin{pmatrix} 2 & 4 & 8 \\ 0 & 4 & 0 \\ 0 & 2 & 0 \\ 4 & 0 & 2 \\ 3 & 5 & 7 \\ \end{pmatrix}$$ ### To solve this, I will: 1. **Calculate the rank** of each matrix by reducing them to row echelon form (REF) or row reduced echelon form (RREF). 2. **Determine the basis for the row space** by identifying the linearly independent rows in the REF/RREF. 3. **Determine the basis for the column space** by identifying the linearly independent columns of the original matrix that correspond to the pivot columns in the REF/RREF. Would you like me to go step-by-step through the solution for any specific matrix, or would you prefer an overall summary of how to approach these types of problems?