Query regarding determining linear dependence and independence in 3x3 matrix

In the lecture Linear dependence and independence (3x3)

At time 4:27, there is a quiz to determine if given matrices are linearly dependent or independent. Now the matrix has the shape:

[ 1 0 1
0 1 0
3 2 3 ]

The solution to this is that since 3Row1 + 2Row2 = Row3, the system is linearly dependant. Adding +2 to Row1 also gives Row3. Is this a fair method to check relationship? If yes then why not use this its much simpler. If no, then why not?

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Hello @i200660_Mirza_Ubaidu,

Interesting thought! :wink:

However, “adding +2” is equivalently adding a new row which is [2 2 2] such that Row 1, Row 3 and the New Row are linearly dependent. Since the question is asking whether the three existing rows are linearly dependent or not, adding a new row isn’t an option, so it is not a valid method to check the relationship.

However, if you claim that your [2 2 2] comes from “(Row 2 + Row 3) / 2”, then it is acceptable, but the key is, whether the “new row” is a linear combination of the existing rows or not. If yes, then OK. If not, then we cannot create that new row.

See? We always have to consider this by linearly combining existing rows and there is no way to get rid of that. Consider only the existing rows are the spirit of discussing whether the existing rows are linearly dependent of each other or not.



@rmwkwok I suppose that makes sense. Thank you.