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September 7th, 2019, 10:38 AM   #1
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Confusion about span in Linear Algebra

Hi there, this is my first post here. I am currently studying some control theory related stuff and its basically linear algebra. However, I have some problems. I have given this exercise on which I already got an answer in Mathematics Stackoverflow:

Quote:
 The controllability matrix $P=\left[\begin{array}{lll}{A^{2} B} & {A B} & {B}\end{array}\right]$ of a state-space model $(A,B,C,D)$ has the following SVD. Determine the kernel and image space of $P$. $$P=\left[\begin{array}{lll}{u_{1}} & {u_{2}} & {u_{3}}\end{array}\right]\left[\begin{array}{ccc}{\sigma_{1}} & {0} & {0} \\ {0} & {\sigma_{2}} & {0} \\ {0} & {0} & {\sigma_{3}}\end{array}\right]\left[\begin{array}{c}{v_{1}^{T}} \\ {v_{2}^{T}} \\ {v_{3}^{T}}\end{array}\right]$$
Quote:
 the vectors $u_1,\dots,u_r$ will form an orthonormal basis for the image of $P$ and the vectors $v_{r+1},\dots,v_{n}$ will form a basis for the kernel of $P$.
However, when checking other sources I found something which does in my eyes not fit the answer above:
$$M=\left[\begin{array}{cc}{U_{1}} & {U_{2}}\end{array}\right]\left[\begin{array}{cc}{\Sigma_{1}} & {0} \\ {0} & {0}\end{array}\right]\left[\begin{array}{c}{V_{1}^{\prime}} \\ {V_{2}^{\prime}}\end{array}\right]$$
$$\text { Nullspace: } \mathcal{N}=\operatorname{span}\left[V_{2}\right]$$
$$\operatorname{image}(M)=\operatorname{span}\left[U_{1}\right]$$

If I got this right the span is the spanning set of a matrix. I see how I get it despite a lack of intuition about it. What I don't get is why the are writing those indices. Isn't $V_2$ a vector? Whats the span of a vector and why the second one? The same for $U$. The answer I received and this definition are somehow similar but not identical. The answer mentions a set of vectors whilst this definition is about the span of a (single) vector. I am a little bit confused.

Thank you guys. I am looking forward to be part of this community Regards, Bob Tags algebra, confusion, linear, span Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post NineDivines Linear Algebra 2 March 8th, 2019 01:17 AM pregunto Linear Algebra 1 January 27th, 2019 06:52 PM zzzhhh Linear Algebra 3 September 26th, 2017 10:00 PM WeMo123 Linear Algebra 1 January 11th, 2011 04:04 PM roncarlston Algebra 9 March 15th, 2009 11:54 AM

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