- #1
aurorasky
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Is it true that any linear map between two arbitrary finite-dimensional vector spaces is continuous? Is it differentiable?
Linear maps between finite dimensional spaces
In summary, it is true that any linear map between two arbitrary finite-dimensional vector spaces is continuous. Differentiability can be determined by finding a linear map that approximates the map well.
- #2
- 22,183
- 3,322
Hi aurorasky! 
This is indeed true! Take an arbitrary linear map [itex]f:\mathbb{R}^n\rightarrow \rightarrow{R}^m[/itex]. Then it suffices that the components f1,...,fm are continuous. And this is true since
[tex]\begin{eqnarray*}
|f_1(x_1,...,x_n)|
& = & |x_1f_i(1,0,...,0)+...+x_nf_i(0,...,0,1))|\\
& \leq & |x_1||f(1,0,...,0)|+...+|x_n||f(0,...,0,1)|\\
& \leq & \|(x_1,...,x_n)\|_\infty(|f(1,0,...,0)|+...+|f(0,...,0,1)|)\\
\end{eqnarray*}[/tex]
which implies continuity. For differentiability, we have to find a linear map u such that
[tex]\lim_{h\rightarrow 0}{\frac{f(x_0+h)-f(x_0)-u(h)}{\|h\|}}=0[/tex]
take u=f and it's easy to see that this is true.
aurorasky said:
This is indeed true! Take an arbitrary linear map [itex]f:\mathbb{R}^n\rightarrow \rightarrow{R}^m[/itex]. Then it suffices that the components f1,...,fm are continuous. And this is true since
[tex]\begin{eqnarray*}
|f_1(x_1,...,x_n)|
& = & |x_1f_i(1,0,...,0)+...+x_nf_i(0,...,0,1))|\\
& \leq & |x_1||f(1,0,...,0)|+...+|x_n||f(0,...,0,1)|\\
& \leq & \|(x_1,...,x_n)\|_\infty(|f(1,0,...,0)|+...+|f(0,...,0,1)|)\\
\end{eqnarray*}[/tex]
which implies continuity. For differentiability, we have to find a linear map u such that
[tex]\lim_{h\rightarrow 0}{\frac{f(x_0+h)-f(x_0)-u(h)}{\|h\|}}=0[/tex]
take u=f and it's easy to see that this is true.
Last edited:
- #3
aurorasky
- 7
- 0
Thanks! Also, if a map is differentiable, it is also continuous and so the proof can be made really simple, right?
BTW, could you explain to me how to type latex code?
BTW, could you explain to me how to type latex code?
- #4
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aurorasky said:
Yes, but the proof that a differentiable map is continuous uses that linear maps are continuous... (at least the proof I've seen)
Click on the LaTeX symbols to see what I did.
- #5
mathwonk
Science Advisor
Homework Helper
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of course you have to give the topology to decide continuity. presumably you meant the norm topology, i.e. the one induced from R^n by the isomorphism given by any basis. In that case a linear map is given by a polynomial hence continuous.
then you have to give the differentiable structure, which again is rpesumably the one defined by a linear isomorphism with R^n. Now of coiurse for it to be meaningful, you need to know that every linear isomorphism defines the same differentiable structure, which is equivalent to your original question for R^n, i,.e is every linear map on R^n differentible,
well by definition a map if differentiable if locally it is approximated very well by a linear map, so duhh,... yes a linear map is differentiable.
more precisely a map f is tangent to zero if |f(x)|/|x|-->0 as x-->0. and a map g is differentiable at a if there is a linear map L such that the map g(a+x)-g(a)-L(x) is tangent to zero. Obviously if g is linear and we take L = g, then
g(a+x)-g(a)-L(x) = 0, hence is certainly tangent to zero.
in infinite dimensions some linear maps are not (norm) continuous, and differentiability may be defined as "approximable by a continuous linear map". is this your confusion? i.e., is some nincompoop teaching you in finite dimensional differential calculus before finite dimensional calculus? (that happened to me.) (I was the nincompoop.)
then you have to give the differentiable structure, which again is rpesumably the one defined by a linear isomorphism with R^n. Now of coiurse for it to be meaningful, you need to know that every linear isomorphism defines the same differentiable structure, which is equivalent to your original question for R^n, i,.e is every linear map on R^n differentible,
well by definition a map if differentiable if locally it is approximated very well by a linear map, so duhh,... yes a linear map is differentiable.
more precisely a map f is tangent to zero if |f(x)|/|x|-->0 as x-->0. and a map g is differentiable at a if there is a linear map L such that the map g(a+x)-g(a)-L(x) is tangent to zero. Obviously if g is linear and we take L = g, then
g(a+x)-g(a)-L(x) = 0, hence is certainly tangent to zero.
in infinite dimensions some linear maps are not (norm) continuous, and differentiability may be defined as "approximable by a continuous linear map". is this your confusion? i.e., is some nincompoop teaching you in finite dimensional differential calculus before finite dimensional calculus? (that happened to me.) (I was the nincompoop.)
FAQ: Linear maps between finite dimensional spaces
1. What is a linear map?
A linear map is a mathematical function that preserves the structure of vector spaces. In other words, it takes in vectors as inputs and produces vectors as outputs, while also satisfying the properties of linearity.
2. What are finite dimensional spaces?
Finite dimensional spaces are vector spaces that have a finite basis. In other words, they have a finite number of linearly independent vectors that can span the entire space.
3. How do linear maps between finite dimensional spaces work?
Linear maps between finite dimensional spaces take vectors from one finite dimensional space and map them to vectors in another finite dimensional space. This is done by applying a linear transformation to the input vectors, resulting in an output vector in the other space.
4. What are some examples of linear maps between finite dimensional spaces?
Some common examples of linear maps between finite dimensional spaces include rotations, reflections, and dilations in Euclidean spaces. In linear algebra, linear transformations such as matrix multiplication and differentiation are also considered linear maps.
5. Why are linear maps between finite dimensional spaces important?
Linear maps between finite dimensional spaces have many applications in mathematics and science. They are essential in solving systems of linear equations, representing geometric transformations, and understanding the behavior of complex systems. They also play a crucial role in fields such as physics, engineering, and computer graphics.