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Let A be a m x n matrix whose columns are vectors of R™. What is the smallest n must be so that these column vectors are a basis of Rm? Explain your answer. Hint: Look at an example with m = 4, n = 3. Every row echelon form for A looks like а 0 d e 0 0 f 0 0 0 where the given letters may or may not be zero. How many vectors must be added or removed to make A's columns into a basis? After answering that question, think more generally where the numbers of rows and columns are variables. The next step of the problem is to explain why it is enough to look at the case of n < m. After you explain this, we can now state how

Let A be a m x n matrix whose columns are vectors of R™. What is the smallest n must be so that these column vectors are a basis of Rm? Explain your answer. Hint: Look at an example with m = 4, n = 3. Every row echelon form for A looks like а 0 d e 0 0 f 0 0 0 where the given letters may or may not be zero. How many vectors must be added or removed to make A's columns into a basis? After answering that question, think more generally where the numbers of rows and columns are variables. The next step of the problem is to explain why it is enough to look at the case of n < m. After you explain this, we can now state how

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Let A be a m x n matrix whose columns are vectors of R". What is the smallest n must be so that these column vectors are a basis of R"? Explain your answer. Hint: Look at an example with m = :4, n = 3. Every row echelon form for A looks like а ь с 0 d e 0 0 f 0 0 0 where the given letters may or may not be zero. How many vectors must be added or removed to make A's columns into a basis? After answering that question, think more generally where the numbers of rows and columns are variables. The next step of the problem is to explain why it is enough to look at the case of n < m. After you explain this, we can now state how many vectors must we add or remove to make this into a basis.
Transcribed Image Text:Let A be a m x n matrix whose columns are vectors of R". What is the smallest n must be so that these column vectors are a basis of R"? Explain your answer. Hint: Look at an example with m = :4, n = 3. Every row echelon form for A looks like а ь с 0 d e 0 0 f 0 0 0 where the given letters may or may not be zero. How many vectors must be added or removed to make A's columns into a basis? After answering that question, think more generally where the numbers of rows and columns are variables. The next step of the problem is to explain why it is enough to look at the case of n < m. After you explain this, we can now state how many vectors must we add or remove to make this into a basis.
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