Lin. transf. and linear independence

eckiller
Messages
41
Reaction score
0
Hi,

If I transform a set of linearly independent vectors by a one-to-one linear transformation, is the transformed set also linearly independent?
 
Physics news on Phys.org
Suppose it isn't. What can you say about it?
 
make sure you know all the definitions of the terms you are using. deciding this problem comes almost immediately just from knowing what the words mean.
 

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 »

Thread 'Decomposition into irreps of compact Lie group'

When decomposing a representation ##\rho## of a finite group ##G## into irreducible representations, we can find the number of times the representation contains a particular irrep ##\rho_0## through the character inner product $$ \langle \chi, \chi_0\rangle = \frac{1}{|G|} \sum_{g\in G} \chi(g) \chi_0(g)^*$$ where ##\chi## and ##\chi_0## are the characters of ##\rho## and ##\rho_0##, respectively. Since all group elements in the same conjugacy class have the same characters, this may be...
View full post »

Similar threads

MHB Does Linear Independence in V Imply the Same in W?
Replies
24
Views
2K
I Characterizing linear independence in terms of span
Replies
3
Views
1K
I Confused about small detail in rank-nullity theorem
Replies
1
Views
1K
MHB Proving $\{w_1, \ldots , w_m\}$ is a Basis of $\text{Lin}(v_1, \ldots , v_k)$
Replies
3
Views
2K
I Wronskian: null space equals the span of independent functions
Replies
23
Views
1K
I Vector subspace and basis vectors in the context of data science
Replies
6
Views
3K
MHB Show that there are vectors to get a basis
Replies
9
Views
2K
I Uniqueness of reduced row echelon form (proof verification)
Replies
4
Views
1K
I Why Use Gram-Schmidt to Make a Unitary Matrix?
Replies
14
Views
4K
I Row space, Column space, Null space & Left null space
Replies
8
Views
3K

Hot Threads

Recent Insights

Back
Top