Proving that Columns are Linearly Dependent

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SUMMARY

The discussion centers on proving that the columns of an m x n matrix A are linearly dependent when m < n. It is established that for linear dependence, the determinant must equal zero or at least one column vector can be expressed as a linear combination of others. The maximum dimension of the space spanned by the columns is m, indicating that with n > m, there are n - m columns that can be represented as linear combinations of the m pivot columns, confirming their linear dependence.

PREREQUISITES
  • Understanding of linear algebra concepts, specifically linear dependence and independence
  • Familiarity with matrix theory, including m x n matrices
  • Knowledge of reduced row echelon form (RREF) and its implications for pivot columns
  • Basic understanding of determinants and their role in linear algebra
NEXT STEPS
  • Study the properties of determinants in relation to linear dependence
  • Learn about reduced row echelon form (RREF) and how to compute it for matrices
  • Explore the concept of pivot columns and their significance in linear algebra
  • Investigate the implications of the Rank-Nullity Theorem in relation to linear dependence
USEFUL FOR

Students studying linear algebra, educators teaching matrix theory, and anyone interested in understanding the fundamentals of linear dependence in vector spaces.

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Homework Statement


Let A be an m x n matrix with m<n. Prove that the columns of A are linearly dependent.

Homework Equations


Its obvious that for the columns to be linearly dependent they must form a determinate that is equal to 0, or if one of the column vectors can be represented by a linear combination of the other vectors.

The Attempt at a Solution


It seems like there has to be more shown to prove this statement, however this is what I have right now:
Let A be an m x n matrix, and let m < n.
Then the set of n column vectors of A are in Rm and must be linearly dependent.

Is this it? or do I need to state a theorem in here somewhere?
 
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B18 said:

Homework Statement


Let A be an m x n matrix with m<n. Prove that the columns of A are linearly dependent.

Homework Equations


Its obvious that for the columns to be linearly dependent they must form a determinate that is equal to 0, or if one of the column vectors can be represented by a linear combination of the other vectors.

The Attempt at a Solution


It seems like there has to be more shown to prove this statement, however this is what I have right now:
Let A be an m x n matrix, and let m < n.
Then the set of n column vectors of A are in Rm and must be linearly dependent.

Is this it? or do I need to state a theorem in here somewhere?

Hint: what is the maximum dimensionality of the space spanned by the columns (regarded as column vectors)?
 
The maximum dimension of the space spanned would have to be m+1, correct? For example if the vectors were from R3 we would need 4 column vectors so that they were linearly dependent.
 
Last edited:
Show that the reduced row echelon form of the mxn matrix will have at most m pivots. Then there are n-m columns without pivots, which can all be expressed as linear combinations of the columns with pivots. Since they are linear combinations of other columns, they are linearly dependent.
 
It would have been helpful if our professor explained what pivots were. Thanks though Izzy I'll make sense of what you explained and go from there.
 
B18 said:
The maximum dimension of the space spanned would have to be m+1, correct? For example if the vectors were from R3 we would need 4 column vectors so that they were linearly dependent.

Are you telling me that you think 4 or more 3-component vectors can possibly span a space of dimension 4?
 

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