W1 PQ1:what operations allowed while determining linear independence or dependence

Course Q&A Math for Machine Learning Linear Algebra for Machine Learning and Data Sc...
1

In the ungraded quiz we see a 2*2 matrix like
[1 2
2 3]
so I selected the matrix to be linearly dependent. since we can derive the second row by adding 1 to each column for row 1, and my answer was wrong.
My questions are:

  1. Why is my assumption and answer wrong?
  2. Is it required to only use * and / operators to determine “dependency” in between 2 rows?
  3. Does the input matrix require to be a 3*3 matrix?
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2

can you just post the screenshot of the question without posting answers to check what you are talking about because you didn’t seem to mention what was the question about!!

3

Thank you for the reply, but wasn’t what i posted straightforward enough? i mean given a 2*2 matrix determine if its there exists linear dependence or independence ?

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4

The operations you are allowed to do are linear combinations of the rows. For example, multiply the first row by -2 and then add the first row to the second row. What happens then? The vector (1, 1) has nothing to do with that matrix, so just adding that is not relevant. Have you gotten to the lecture where he talks about “row echelon form” yet?

The notion of linear independence of the rows of a matrix applies to square matrices of any size.

The more advanced way to tell if the matrix is full rank or not is to use the determinant, but I’m not sure from your question whether you’ve gotten to that lecture yet or not.

But the higher level point is that all this is covered in the lectures in some detail. It might be worth watching them again.:nerd_face:

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5

That operation is not a linear combination of the rows. You’ve invented a new row [1, 1], and thrown that into the mix.

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6

I see, but if that were the case nowhere was it mentioned in the lecture that “applying some operations to a row” ==> creates a new row? if that were the case take the example below
[1 1
0 0
2 2]
Now, can I say

  1. row3 can be derived by multiplying row1 by 2 and adding row2?
  2. If yes,then by what you wrote about “inventing a new row” does that mean for each operation I applied from pt.1 above I end up creating new rows? Im guessing no.

Thats where im having issues.

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7

Thank you for your reply, but my question isn’t at all referring to usage of determinants nor am I at the stage of applying “row echelon form”.
Also, i do not understand your comment

For example, multiply the first row by -2 and then add the first row to the second row.

are you referring to my example or something else?

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8

nevermind i think i found out the issue.

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9

Maybe you have not gotten to it yet, but please watch this lecture. In that he shows lots of examples of what Tom and I mean by “linear combinations of rows” and how that is applied.

Note that it does not involve adding new rows to the matrix.

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