03 Linear Dependence, Span, Basis

Linearly-Dependent/Independent¶

Let \(\alpha_1 u_1 + \alpha_2 u_2 + \dots + \alpha_n u_n = \vec 0\)

Working¶

1. Column-wise

2. Condition	Solution	Conclusion
   \(r(A) = n\)	unique \((0, 0, \dots)\)	independent
   else	infinitely-many	dependent

Span¶

Let \(\vec v = \alpha_1 u_1 + \alpha_2 u_2 + \dots + \alpha_n u_n\)

Working¶

1. Column-wise

2. Condition	Solution	Conclusion
   \(r(A) = r(A:B) = n\)	unique	span
   \(r(A) = r(A:B) < n\)	infinite	span
   else		not span

Basis¶

1. \(S\) is Linearly-independent –> row-wise working
2. \(S\) spans \(V\) –> dim(\(V\)) = no of vectors in \(S\)

Note: \(\vec 0\) has no basis

Dimension¶

no of unknowns

no of vectors in its basis

dim \((\vec 0)= 0\)

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