[Linear Algebra] Matrix-Vector Equation Ax=b
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- เผยแพร่เมื่อ 5 ก.พ. 2025
- We learn how to solve the matrix equation Ax=b.
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We discuss the matrix-vector equation Ax=b.
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![[Linear Algebra] Computing Ax and The Identity Matrix](http://i.ytimg.com/vi/wVCSH9lthYo/mqdefault.jpg)
![[Linear Algebra] Computing Ax and The Identity Matrix](/img/tr.png)
If anyone wishes to understand Linear Algebra go and watch Essence of Linear Algebra and then come back here to see how problems are solved. Things become very clear after this.
Yep. That's what I'm currently doing.
this is the way.
I would advise against this. Go watch the playlist after you know the material.
Thanks so much! My teacher did a good job of explaining this, but I couldn't quite wrap my head around it like other students did. Having it explained in a different way helped!
Alright, but what if the matrix is the unknown element? I am currently studying symmetries and I often get cases like C(1,2,3)=(2,1,3) where C is some operation. How would I find the matrix C which facilitates this transformation?
I knew how to do dot product and I did a lot of complicated matrices in my mathematical physics class, but I didn’t know why until now!
At 6:03 I did it the other way round. Is it still correct?
Very useful man! Good job!!
n lives in m is the reason that you might have a vector with 6 places and end up with 4 or however many rows you have in your matrix.
Thank u very much, really i found it helpful💛.
Probably just a weird graphical coincidence - but is there anything to the analogy that the coloumns and rows being orthogonal directions, produce the same as the n (dot) x = n (dot) p rule for deriving general equations from the norm and direction-to-point vectors?
See you understand this to a point I have no idea what you even said like that could've been random sci fi lingo and I wouldn't of known.
hi i love you for god.
What about ax+b=0?
why didn''t you reduce to pivot position in the last example
Why did you add row 2 to row 3?
To solve it completely
After adding row 2 to row 3 , we can easily get one variable solved
@@nehasihag_a_0307 but you could have used subtraction why adding
it seems so far that linear algebra is just having different funny ways to say "put into augmented matrix, convert to RREF, solve"
This is a MCQ type question Tick the correct answer...:-
for the given set of simultaneous equations
1.5x -0.5y =2
4x+2y+3z=10
7x+y+5z=10
choose correct one of the following:-
1) solution is unique
2) infinitely many solution exist
3) Equation are incompatible
4) finite number of multiple solution exist.
please answer this and explaination also. i.e. PLEASE SOLVE .
You do realise you could subsitute equation 1 into the other equations and just rearrange or just put them in a matrix of ...
[1.5 - 0.5 0 ] [ x ] [ 2 ]
[4 2 3] X [ y ] = [10]
[7 1 5] [ z ] [10]
Which is in the form of this video, I'll leave the rest to you. Good luck
How is a vector defined?
A vector is an undefined term, so it is not explicitly defined, It is however implicit in the properties of a vector space.
However, many people will "define" it by some of its characters. In other words they don't really define it, but they characterize it. for example some will characterize a vector by saying that a vector is an object that has a direction and a magnitude.
In the last example. That is the only solution right? There is only one unique solution? Thank you.
MasayoMusic that is the only correct solution my friend. Check your math, prob a minor error.
Why didn’t you reduce the last matrix to REF?
Was thinking the same thing! Thanks David McMichael for clarifying :)
Because it was unnecessary
@@redshiftedlight205 why is it unnecessary
stuck at 8:37. why did you not reduce to REF
As you saw, it's not necessary, but you could, which is what I would also do. You would end up with:
│x₁ x₂ x₃│ b │
│1 0 0 │-11│
│0 1 0 │ 3│
│0 0 1 │ 0│
So:
x₁ = -11
x₂ = 3
x₃ = 0
Which is the same as in the video.
NICE video
How did u deduce the final results from the matrix? I.e 2x =6
Look at the last row of the matrix. What would that look like if you wrote it as an equation?
That's my question I'm
Confused
@@francisjuarez7342 It's just like solving for x in algebra. 2x=6 -> x=3 when you divide both sides by 2.
Strictly speaking
[ *a1* ... *an* ] . [ x1]
[ x2 ]
.
.
.
[xn]
is a slight abuse of notation using a dot product.
Dot products are used to take product of two lists (not specifically using matrices).
But this still works as a heuristic , it is valid.
In general a dot product operates on two lists of numbers.
en.wikipedia.org/wiki/Dot_product#Algebraic_definition
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