MHB How Do Basis and Vectors Work in Linear Algebra?

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Basis and vectors in linear algebra are defined through the concept of coordinates. To express a vector v in terms of a basis, one can use a linear combination of basis vectors, as shown in the example where v is represented as a combination of three vectors. The coordinates of v in the basis B can be found by solving the corresponding system of equations. Additionally, another vector w can be expressed as a linear combination of the basis vectors with specific coefficients. Understanding these concepts is essential for working with vector spaces in linear algebra.
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Hi guys,

I'm back and have another Linear Albgera question!

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Thanks in advance.

No idea how to start
 
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In both cases, you only need to apply the definition of coordinates. For $(a)$ express $v=x_1(1,0,1)+x_2(1,1,0)+x_3(0,1,1)$, solve the system and $[v]_{\mathcal{B}}=(x_1,x_2,x_3)^T$. For $(b)$, $w=1v_1+2v_2-3v_3$, where $\mathcal{B}=\{v_1,v_2,v_3\}$. Let us see what do you obtain.
 
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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