Science:Math Exam Resources/Courses/MATH221/December 2008/Question 01 (a)
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Question 01 (a) 

Let denote the matrix Find a basis for the column space of . 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

A set of vectors is a basis if they span the entire vector space and are linearly independent. 
Hint 2 

The rows of a matrix are linear independent if and only if the determinante of the matrix is not equal to . The columns of a matrix are linear independent if and only if the determinante of the matrix is not equal to . 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The column space of is The set of vectors spanning the column space of form a basis if it is linearly independent. We can check this by determining the determinante of . If , then the columns of are linearly independent.
Hence, the columns of are linear dependent. Indeed, we see that the third column can be expressed as , while the set of the first two column vectors of is linearly independent. Therefore the basis of the column space of matrix is . 