MHB Proving Determinant: u,v in R^n | Det(I + uv^T) = 1 + v^Tu

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The discussion focuses on proving the determinant identity Det(I + uv^T) = 1 + v^T u for vectors u and v in R^n. It highlights that this identity is a special case of the Matrix Determinant Lemma. Participants are encouraged to explore the properties of determinants and matrix operations to derive the proof. The relationship between the identity matrix and the outer product of vectors is emphasized as a key aspect of the proof. Understanding this determinant identity is crucial for applications in linear algebra and related fields.
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\[ \text{Let u,v } \in \mathbb{R}^n \;\; \text{Show that } \;\;, Det(I + uv^T) = 1 + v^T u \]

I is the identity matrix nxn
any hints ?
 
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Thread 'How to define a vector field?'

Thread 'How to define a vector field?'

Hello! In one book I saw that function ##V## of 3 variables ##V_x, V_y, V_z## (vector field in 3D) can be decomposed in a Taylor series without higher-order terms (partial derivative of second power and higher) at point ##(0,0,0)## such way: I think so: higher-order terms can be neglected because partial derivative of second power and higher are equal to 0. Is this true? And how to define vector field correctly for this case? (In the book I found nothing and my attempt was wrong...
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