Kernel and Range: Understanding Linear Transformation in Algebra

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The kernel and range of a linear transformation are crucial concepts in linear algebra, as they provide insights into the solutions of equations related to the transformation. The kernel represents all solutions to the equation Ax = 0, while the range identifies all possible outputs b for which Ax = b has a solution. Understanding these concepts is essential for grasping the broader implications of linear transformations in both linear and abstract algebra. Although they may initially seem challenging, mastering the kernel and range is beneficial for deeper mathematical comprehension. Ultimately, these concepts are fundamental to the study and application of linear algebra.
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Why are we interested in looking at the kernel and range (image) of a linear transformation on a linear algebra course?
 
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If we are at all interested in a linear transformation, wouldn't we want to know all we could about it? Do you remember, in basic algebra, solving equations a lot? Same thing here. Finding the kernel of a linear transformation, A, is the same as findig all solutions to Ax= 0. Finding the image is the same as finding all b such that Ax= b has a solution.
 
HallsofIvy has made the vital point clear. For the OP, I just want to mention that the concepts of kernel and range are vitally important to all of linear algebra and (later) abstract algebra. They may seem unintuitive at first, but it is worth the effort...
 
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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