Linear independence and dependence

yoyo
Messages
21
Reaction score
0
Hi everyone, having problems with this question, can anyone please help

Question: consider a 2 x 2 matrix, can you construct a matrix whose columns are linearly dependent and whose rows are linearly independent?

My answer is no. I cannot think of any combination that would make this true? Or is there? If so why?

Thanks
 
Physics news on Phys.org
No, because having linearly dependent columns gives a singular matrix yet linearly independent rows gives a non-singular matrix.
 
Thanks, make sense.
 
just do the algebra.
 

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

I Characterizing linear independence in terms of span
Replies
3
Views
1K
I Uniqueness of reduced row echelon form (proof verification)
Replies
4
Views
1K
I Row space, Column space, Null space & Left null space
Replies
8
Views
3K
I Vector subspace and basis vectors in the context of data science
Replies
6
Views
3K
I Dimension and solution to matrix-vector product
Replies
4
Views
3K
A linear programming, polyhedron and extreme points
Replies
3
Views
478
I Wronskian: null space equals the span of independent functions
Replies
23
Views
1K
I Confused about small detail in rank-nullity theorem
Replies
1
Views
1K
I Proving a set is linearly independant
Replies
5
Views
2K
I Proving linear independence of two functions in a vector space
Replies
5
Views
2K

Hot Threads

Recent Insights

Back
Top