Rank of a Matrix

[pine spoke]

I tried manipulating, for Row Echelon form.
But I am getting confused in between

[lilac eagle]

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[tight tapir]

could you explain what a rank of a matrix is?

[pine spoke]

The rank of a matrix is the maximum number of linearly independent rows or columns in that matrix

[tight tapir]

okay

from there we know that b=2 is a trivial solution

now to tackle b=-6

[pine spoke]

I think, first, I need to get a clear idea of what is actually meant by the rank of a Matrix.

[thin moth]

Do you understand what linear independence is?

[thin moth]

What did you get as you turned the matrix into row-echelon form?

This is important

[sick ingot]

you could view it as dimension of the subspace spanned by rows/columns (why these dimensions are same isn't trivial)

[pine spoke]

Yes, Rank of a matrix is actually the dimension of a vector space generated or spanned by its columns..

[pine spoke]

Yes, For example, if we take this Matrix , the third column is linearly dependent on the first two, so we can say that the rank should be less than 3, I.e. "2", in this case.
The first two columns are linearly independent.

@thin moth

@sick ingot

[thin moth]

You seem to have done it correctly

[pine spoke]

You see, in my college

[thin moth]

Indeed, you show with the row echelon form that if b is not 2, and not -6, then the rank is full

So you can check the remaining 2 cases individually

[pine spoke]

They defined "Rank" with respect to just "Pivots"

So I let myself onto a journey to actually understand Rank

And now I am clear, Thanks to you all too 😊

[pine spoke]

Yes Indeed

[thin moth]

Hm I guess you got the help you need already

Then please close the post if you are done

Or feel free to ask if you have more questions

[pine spoke]

Yes, I want to keep this open to because I am still reading about Ranks and Vector Spaces

If any doubts, I will ask here

[thin moth]

I don't think it will be deleted if you close it (as long as you keep the link to it)

But sure, I don't mind

You can keep it open

[pine spoke]

Yes Okay Sure 👍

Thanks

[wild cedar]

@pine spoke

<:HelpIcon:1304095958283321385>| Help Reminder

Hello thylegend_, this is a friendly reminder that your thread has been inactive for more than 24 hours. If you no longer need assistance, please consider closing the thread using the +close command.

[pine spoke]

+close

[thin moth]

Seems like the bot is down, please try again in a couple of hours

[pine spoke]

Oh Okay, I will

[thin moth]

@pine spoke Try again

[pine spoke]

+close

[thin moth]

Urgh crap

Sorry about that

our bot is pinging a lot of people to close their threads (this is intended)

[pine spoke]

np, sometimes happens

[thin moth]

so it got quarantined

we are asking the server manager to do something about it

[pine spoke]

ohh that's a burden

[thin moth]

Yeah it's just that we want people to clean up after themselves

or they'll keep getting annoyed by the bot

[pine spoke]

ok it's completely understandable

[pine spoke]

lol yeah

[thin moth]

I'll let you know when the bot is online again

Once again, sorry for the bother

[honest sinew]

+close