6.5 Theorem of Vector List Size m x 1

#Type/Theorem #Topic/Linear_Algebra

Types: N/A
Examples: 6.5.1 Example of Linear Dependent Set
Constructions: N/A
Generalizations: 6.4 Linearly Dependent Vectors

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Theorem

Any list of $n$ vectors of size $m \times 1$ with $n > m$ is linearly dependent.

Remark. In a linearly dependent set of vectors, at least one vector can be created from the other vectors in the set via linear combination.
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Remark. Linear dependence is connected to the study of the range of the 10 Matrix-Vector Multiplication problem. Given a set of vectors ${\vec{a}}_{1}, {\vec{a}}_{2}, \dots, {\vec{a}}_{n} \in R^{m}$ , we can take the span of that set. Let $A$ be a matrix with $m$ rows and $n$ columns given by

A = [\begin{matrix} {\vec{a}}_{1} & {\vec{a}}_{2} & \dots & {\vec{a}}_{n} \end{matrix}] .

The span of these vectors is the range of the matrix vector multiplication problem. A set of vectors is linearly dependent if we can delete one of the columns of our vector and produce the same range from the reduced matrix.
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Remark. In a linearly dependent set of vectors, at least one vector can be created from the other vectors in the set via linear combination. ⧫

Remark. In a linearly dependent set of vectors, at least one vector can be created from the other vectors in the set via linear combination.
$⧫$