MHB Linear Algebra: Kernel & Range of Linear Transformation

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The kernel and range of a linear transformation are crucial in linear algebra for understanding the properties of linear maps. Specifically, the kernel helps determine if a linear map is injective, as a kernel of only the zero vector indicates injectivity. Additionally, the relationship between the quotient space V/kernel(f) and the image of the transformation provides a practical way to analyze linear maps. These concepts simplify complex problems by allowing focus on the image rather than the quotient space. Understanding these aspects is essential for deeper insights into linear transformations.
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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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They can be useful. Suppose you have a linear map f: V \to W. If you want to know if this linear map is injective (i.e one-to-one map) then you can take a look at the kernel: $$\ker( f)=\{0\} \Leftrightarrow \ f \ \mbox{is injective}$$

There's also the following result.
$$V/\mbox{ker}( f) \cong \mbox{Im}( f)$$
which can be very useful because it's easier to work with the image in stead of the quotientspace $V/\mbox{ker}( f)$

These are offcourse a lot of other results but these two are the first I could remember immediately.
 
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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