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April 20

Rank of a Wide Matrix

If a matrix A has m rows and n columns such that m < n, does that necessarily imply that Rank(A) ≤ m? I believe the answer is yes because the column vectors must be linearly dependent, and visually, it seems clear that it's not possible for all the columns in RREF(A) to have a leading 1, but I just want to confirm that I'm not making a mistake. Puzzledvegetable^{Is it teatime already?} 02:15, 20 April 2023 (UTC)[reply]

Yes, see for instance here. Using the concept of row rank, it is obvious that it is at most the number of rows. Since rank = column rank = row rank (see § Proofs that column rank = row rank), the result follows. --Lambiam 07:00, 20 April 2023 (UTC)[reply]

I Do Not Understand Negative Numbers

Please explain negative numbers to me as for my entire life the logic in performing arithmetic with negative numbers has eluded me. When you add a negative number it goes in the positive direction because the positive direction is a fixed concept/direction and the addition symbol represents that. When you subtract negative numbers it goes in the negative direction because the negative direction is a fixed concept/direction and the subtraction symbol represents that. When you multiply a negative number it goes in the negative direction because it just multiplies the numbers, adding to it in the direction it is already headed. When you divide negative numbers it halves it thus bringing it in the positive direction. What i simply cannot understand is hy when there a;pears to be two minuses in an arithmetic equation the negative cancels out whatever other value is continuously without even counting the values between which of course would devalue value y please help me understand why it is done like this it doesn't seem like this lines up with reality either.216.168.139.239 (talk) 216.168.139.239 (talk) 14:58, 20 April 2023 (UTC)[reply]

When I was four years old my daddy introduced me to the concept like this: if you walk -2 blocks south, you go +2 blocks north; if you walk -2 blocks north, you go +2 blocks south. South is the negative of north and vice versa. The opposite of the opposite of anything is the original thing. — A few years later I mistakenly thought, by analogy, that dividing by a negative number would be the same as multiplying by the positive … —Tamfang (talk) 15:57, 20 April 2023 (UTC)[reply]

Can you rewrite this? i keep rereading it and i'm not internalizing a thing! ADHD :p 216.168.139.239 (talk) 16:01, 20 April 2023 (UTC)[reply]

There's a lot of philosophical issues that have been involved in people "understanding" negative numbers. Negative number#History explains a lot of this, but ultimately you don't need to worry about any of that if you just memorize the algorithms for working with negative numbers. Shut up and calculate, and just follow the rules for solving the problems. Some of your explanations above are confusing and wrong, so let me explain them the best I can.

When you add a negative number to a positive number, that's the same as subtraction. Rearrange the equation as a subtraction problem, and solve it that way, subtracting the negative from the positive. So if you need to solve -3 + 4, just rewrite it as 4 - 3 = 1.
When you add a negative number to a negative number, just add them together and put the negative out front. So, if you need to solve -3 + -4, just rewrite it as -(3+4) = -7.
When you multiply or divide, you only need to keep track of parity. What that means is that you just need to keep track of the total number of negative signs in the calculation, and if it's odd, make the answer negative, and if it's even, make the answer positive. So something messy like (13 x -11)/(-7 x -12), you would just solve it without the negative signs as (13 x 11)/(7 x 12), and since we have three total negative signs (an odd number of them) you put a negative sign in front of your answer.
For exponents, a negative power means perform the operation 1/(whatever) to answer when you're done. Thus, something like 2^-3 is the same as 1/(2³) or 1/8.
For roots, it's the same as the exponent rule, a negative root means do 1/answer of what's under the radical symbol, thus something like ^-3√27 = 1/³√27 = 1/3.

That should cover all of the rules for solving math involving negative numbers. You just need to memorize those rules, and apply. It's also possible to learn why you do each of them, but understanding why they each work isn't necessary in getting the correct answers. Just learn the rules, and do them every time, and you'll always get the correct answer. --Jayron 32 16:18, 20 April 2023 (UTC)[reply]

For some reason most people seem to understand money ;-) Just think of negative money as a debt you have to pay and then work out how much you have overall when adding a debt to your existing amount. Multiplying a debt by a positive amount increases the debt. Taking away times a debt reduces it and you can end up in the money overall. NadVolum (talk) 17:34, 20 April 2023 (UTC)[reply]

If may help to read the article Negative number and ask about anything you don't understand. Especially read the history section; negative numbers were not accepted right away, mainly because people had a hard time interpreting them. --RDBury (talk) 18:06, 20 April 2023 (UTC)[reply]

Another way of thinking about numbers is that they are like floor levels. Imagine a really tall skyscraper. Like in Europe, the ground floor level is 0. One floor up is level 1, the next is 2, and so on. But the building also has many underground levels. The level just below the ground level is level −1, that below it is −2, and so on. There is an elevator, which has unusual buttons. Instead of pressing the button for the floor you want to go to, you have to tell the elevator how many floors you want to go up or down. To go up three floors, you need to press the button that has +3 written on it. If you are at level 2 and you press that button, the elevator goes up three levels and you are at level 5. But now suppose you are at level 2 and you want to go to level −1. That is three floors down. Then you must press the button that has −3 written on it. --Lambiam 18:41, 20 April 2023 (UTC)[reply]

what? 216.168.139.240 (talk) 216.168.139.240 (talk) 15:35, 25 April 2023 (UTC)[reply]