Non-square matrix and linear independence

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SUMMARY

A non-square matrix can indeed have linearly independent columns, provided it has more rows than columns. For example, the matrix \[\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}\] demonstrates this concept with two columns and three rows. Conversely, if a matrix has more columns than rows, the columns cannot be linearly independent, as the number of vectors exceeds the dimensionality of the space.

PREREQUISITES
  • Understanding of linear algebra concepts, specifically linear independence
  • Familiarity with matrix dimensions and properties
  • Knowledge of vector spaces and their dimensionality
  • Basic understanding of determinants and their role in linear independence
NEXT STEPS
  • Study the concept of linear independence in greater detail
  • Explore the implications of matrix rank and its relationship to linear independence
  • Learn about the properties of non-square matrices in linear transformations
  • Investigate the use of the Reduced Row Echelon Form (RREF) to determine linear independence
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Students and professionals in mathematics, particularly those studying linear algebra, as well as educators seeking to clarify concepts related to matrix theory and linear independence.

torquerotates
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Can a non square matrix have linearly independent columns? I can't take the determinant so I can't tell.
 
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torquerotates said:
Can a non square matrix have linearly independent columns? I can't take the determinant so I can't tell.

Yes. For instance,
\left[\begin{array}{cc} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{array}\right]
Of course it will have to have more rows than columns.
 
If, on the other hand, the matrix has more columns than rows, the columns cannot be independent. If there are say, m columns and n rows, with m> n, then the columns are n dimensional vectors and a set of m vectors of n dimensions, with m> n, cannot be independent.
 

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