MHB How Do Basis and Vectors Work in Linear Algebra?

  • Thread starter Thread starter Kaspelek
  • Start date Start date
  • Tags Tags
    Basis Vectors
Kaspelek
Messages
26
Reaction score
0
Hi guys,

I'm back and have another Linear Albgera question!

2zp6oed.png

​

Thanks in advance.

No idea how to start
 
Last edited by a moderator:
Physics news on Phys.org
In both cases, you only need to apply the definition of coordinates. For $(a)$ express $v=x_1(1,0,1)+x_2(1,1,0)+x_3(0,1,1)$, solve the system and $[v]_{\mathcal{B}}=(x_1,x_2,x_3)^T$. For $(b)$, $w=1v_1+2v_2-3v_3$, where $\mathcal{B}=\{v_1,v_2,v_3\}$. Let us see what do you obtain.
 
Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'

Thread 'Determine whether ##125## is a unit in ##\mathbb{Z_471}##'

This is the question, I understand the concept, in ##\mathbb{Z_n}## an element is a is a unit if and only if gcd( a,n) =1. My understanding of backwards substitution, ... i have using Euclidean algorithm, ##471 = 3⋅121 + 108## ##121 = 1⋅108 + 13## ##108 =8⋅13+4## ##13=3⋅4+1## ##4=4⋅1+0## using back-substitution, ##1=13-3⋅4## ##=(121-1⋅108)-3(108-8⋅13)## ... ##= 121-(471-3⋅121)-3⋅471+9⋅121+24⋅121-24(471-3⋅121## ##=121-471+3⋅121-3⋅471+9⋅121+24⋅121-24⋅471+72⋅121##...
View full post »

Thread 'Trouble understanding an online solution to an exercise in Dummit & Foote'

##\textbf{Exercise 10}:## I came across the following solution online: Questions: 1. When the author states in "that ring (not sure if he is referring to ##R## or ##R/\mathfrak{p}##, but I am guessing the later) ##x_n x_{n+1}=0## for all odd $n$ and ##x_{n+1}## is invertible, so that ##x_n=0##" 2. How does ##x_nx_{n+1}=0## implies that ##x_{n+1}## is invertible and ##x_n=0##. I mean if the quotient ring ##R/\mathfrak{p}## is an integral domain, and ##x_{n+1}## is invertible then...
View full post »

Thread 'Questions about non existence of GCDs vs (coimages, cokernels)'

The following are taken from the two sources, 1) from this online page and the book An Introduction to Module Theory by: Ibrahim Assem, Flavio U. Coelho. In the Abelian Categories chapter in the module theory text on page 157, right after presenting IV.2.21 Definition, the authors states "Image and coimage may or may not exist, but if they do, then they are unique up to isomorphism (because so are kernels and cokernels). Also in the reference url page above, the authors present two...
View full post »

Similar threads

I Standard basis and other bases...
Replies
9
Views
3K
I Non orthogonal basis and the lines of its coordinate grid
Replies
9
Views
3K
I Confused about small detail in rank-nullity theorem
Replies
1
Views
1K
I S is set of all vectors of form (x,y,z) such that x=y or x =z. Basis?
Replies
5
Views
1K
I Showing a set is a basis for a vector space
Replies
2
Views
2K
B What unique contributions to math did linear algebra make?
Replies
19
Views
3K
I Question from a proof in Axler 2nd Ed, 'Linear Algebra Done Right'
Replies
3
Views
2K
A How to visualise complex vector spaces of dimension 2 and above
Replies
4
Views
3K
I Applying change of basis to vector b_1 doesn't give b'_1?
Replies
8
Views
2K
A Vector space (no topology) basis
Replies
38
Views
6K

Hot Threads

Recent Insights

Back
Top