Need Explination (linear Indpendance)

  • Thread starter Thread starter stuckie27
  • Start date Start date
Click For Summary
If the last column of the product AB is entirely zero while matrix B has no zero columns, it indicates that the columns of matrix A are linearly dependent. This is because a linear combination of the columns of A can produce the zero vector, which is reflected in the zero column of AB. The entries in each column of AB represent linear combinations of the columns of A, confirming their dependence. The discussion also touches on the nature of A and B as matrices, clarifying that they are not contravariant vectors and that AB is not their inner product. Understanding linear dependence is crucial for interpreting the relationship between the matrices involved.
stuckie27
Messages
12
Reaction score
0
A and B are both Matricies,

Suppose the last column of AB is entirely zero but B itself has no column of zeros. What can be said about the columns of A?

Answer: The columns of A are Linearly Dependant.

Question: Why?
 
Last edited:
Physics news on Phys.org
stuckie27 said:
Suppose the last column o AB is entirely zero but B itself has no column of zeros. What can be said about the columns of A?

Answer: The columns of A are Linearly Dependant.

Question: Why?

Are A and B contravariant vecotrs? is AB their inner (scalar) product? If so, I'm not sure how it can have columns, please clarify.
 
edit, A and B are each a different Matrix.
 
take a column in AB, what do the entries represent? are they in some way related to linear combinations of the columns of A? (yes they are, that isn't rhetorical) and a column of zeroes might mean that... fill in the blanks using the definition of linear (in)dependence.
 
Thread 'How to define a vector field?'

Thread 'How to define a vector field?'

Hello! In one book I saw that function ##V## of 3 variables ##V_x, V_y, V_z## (vector field in 3D) can be decomposed in a Taylor series without higher-order terms (partial derivative of second power and higher) at point ##(0,0,0)## such way: I think so: higher-order terms can be neglected because partial derivative of second power and higher are equal to 0. Is this true? And how to define vector field correctly for this case? (In the book I found nothing and my attempt was wrong...
View full post »

Similar threads

Undergrad Uniqueness of reduced row echelon form (proof verification)
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Did I figure-out z-translation of 2D-coordinates correctly?
  • · Replies 4 ·
Replies
4
Views
2K
High School Array Representation Of A General Tensor Question
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Gaussian elimination for homogeneous linear systems
  • · Replies 26 ·
Replies
26
Views
5K
MHB If 2 columns are identical is there infinite solutions
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Proof concerning the Four Fundamental Spaces
  • · Replies 14 ·
Replies
14
Views
3K
High School Is it invalid to redefine the sgn function in this way in a proof?
  • · Replies 15 ·
Replies
15
Views
5K
MHB How can I find out if this matrix A's columns are linearly independent?
  • · Replies 1 ·
Replies
1
Views
1K
Question about Cyclical Matrices and Coplanarity of Vectors
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Passive Transformation and Rotation Matrix
  • · Replies 1 ·
Replies
1
Views
3K