What is the rank of a given matrix?

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Homework Help Overview

The discussion revolves around determining the rank of a given matrix, specifically the matrix with elements [3 -6], [5 -10], and [-2 4]. The concept of matrix rank and its implications in linear algebra are central to the problem.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants express confusion regarding the notation used for rank, particularly the meaning of "r" in rk(A). Some seek clarification on the definition of rank and its relation to linear independence of columns.

Discussion Status

There is an ongoing exploration of the notation and definitions related to matrix rank. Some participants have provided insights into the standard notation for functions and variables, while others are questioning the clarity of definitions presented in their resources.

Contextual Notes

Participants note potential confusion arising from the notation and definitions in their textbooks, particularly regarding the distinction between scalar variables and functions. The discussion reflects a need for clearer understanding of the terms involved.

g.lemaitre
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Homework Statement



find rk(A) for the following matrix

[3 -6]
[5 -10]
[-2 4]

Homework Equations


The Attempt at a Solution



How am I supposed to find the answer when I don't know what r is? I thought r had to be a number or a scalar and you multiply the whole matrix by it.
 
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g.lemaitre said:

Homework Statement



find rk(A) for the following matrix

[3 -6]
[5 -10]
[-2 4]

Homework Equations


The Attempt at a Solution



How am I supposed to find the answer when I don't know what r is? I thought r had to be a number or a scalar and you multiply the whole matrix by it.

rk(A) is the rank of A, which is the dimension of the image of the matrix, or, equivalently, the number of linearly independent columns in the matrix.
 
ok, got it, but you can easily see how that could throw one off when given the following theorem:

Screenshot2012-07-19at23441AM.png
 
g.lemaitre said:
ok, got it, but you can easily see how that could throw one off when given the following theorem:

Screenshot2012-07-19at23441AM.png

Not really. Didn't your book define rk(A)? The theorem was about dot products.
 
A point that might help here as far as notation goes: usually, functions like sine, cosine, and rank are written in fully upright, non-italic, non-bold letters, e.g. [itex]\sin \theta[/itex] or [itex]\text{rk}(A)[/itex]. Scalar variables, on the other hand, will usually be italicized. [itex]rk[/itex] is the product of the variables [itex]r[/itex] and [itex]k[/itex].
 
Robert1986 said:
Not really. Didn't your book define rk(A)? The theorem was about dot products.

When you're coming across new notation it's easy to get them confused.
 

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