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Let $A:E\rightarrow F$ be a Linear Transformation between finite dimensional vector spaces, with $\mathrm{Rank}(A)=r$ and $\dim E=n$, $\dim F=m$. Prove that there are basis in $E$ and $F$ such that the matrix of $A$ has $a_{11}=\cdots=a_{rr}=1$ and $a_{ij}=0$ everywhere else, as entries.

I thought in the change of basis $ap=qa'$ where $a'$ would be the matrix we want but as I got no information about $a$, $p$, $q$ then this is not a way out definetely. Then as the rank is the maximum number of independent columns and rows I thought I could just erase the ones that are linear dependent but this doesn't guarantee me that the transformation would be the same transformation without the deleted linear dependent columns and rows.

A hint would be apreciated, Thanks in advance.

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If you want to kill a fly with a sledgehammer, you could apply the Singular Value Decomposition. – wildildildlife Jun 25 '11 at 16:35
up vote 3 down vote accepted

The columns of the matrix $A$ are the components of the images of the basis vectors of $E$, i.e. $A(u_i)= A\cdot u_i$ for some basis $<u_1,...,u_n>=E$. Now the null space $\ker A=\{v\in E|\, A(v)=0\in F\}$ is a vector subspace of $E$ and the image space $\text{im} A=\{w\in F\, |\,\exists v\in E, w=A(v)\}$ a subspace of $F$ with both satisfying the Rank-Nullity theorem $\text{rank}(A)+\text{nullity}(A)=n$, that is $$ \dim(\text{im}\,A)+\dim(\ker A) = \dim E $$

This means $E$ is decomposed into the space of vectors which map to $0$ in $F$ through $A$, that is $\ker A$, and its complementary subspace $E-\ker A=:\widetilde{\text{im}} A < E$, the space of vectors which map to all the nontrivial vectors into the image of $A$, i.e. $E=\widetilde{\text{im}} A\oplus \ker A$. In fact this is just another way of stating the first isomorphism theorem for vector spaces (where $<$ means vector subspace) $$ E>\widetilde{\text{im}} A\cong \frac{E}{\ker A}\cong \text{im}\,A<F. $$ With this decomposition of $E$, you choose a basis of $E$ formed by the $r$ basis vectors $\{u_1,...,u_r\}$ of $\widetilde{\text{im}} A$ and the $n-r$ basis vectors $\{u_{r+1},...,u_n\}$ of $\ker A$. In this basis the matrix of $A$ whose columns are the components of the vectors $A(u_i)$, has zeros in the last $n-r$ columns since they are images of vectors in the null space. Finally the first $r$ columns are $A(u_1)...A(u_r)$, but since $A$ has rank $r$ these must be linearly independent and thus a basis of $\text{im}\, A$. Choose this set of $\{A(u_1),...,A(u_r)\}$ as new basis of the image, so that you can complete it to a basis of $F$ adding some $\{v_1,..., v_{m-r}\}\subset F$. Therefore you have arrived to basis of $E$ and $F$ where the decompositions in $F$ of the linear transformation of the $E$ basis are $$ \begin{aligned} & A(u_i)=\sum_{j=1}^r \delta_{ij}A(u_j)+\sum_{j=1}^{m-r} 0\cdot v_j\;\;\text{for } u_i\in\widetilde{\text{im}} A \\ & A(u_k)=\sum_{j=1}^r 0\cdot A(u_j)+\sum_{j=1}^{m-r} 0\cdot v_j\;\;\text{for } u_k\in\ker A \end{aligned} $$ gives you the matrix of $A$ as $$ A=\begin{pmatrix} 1 & 0 & \cdots & 0 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & 0 & \vdots & \ddots & 0 \\ 0 & 0 & \cdots & 1 & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0 & 0 & \cdots & 0 \\ \vdots & \vdots & \ddots & 0 & 0 & \ddots & 0 \\ 0 & 0 & \cdots & 0 & 0 & \cdots & 0 \\ \end{pmatrix}= \begin{pmatrix} \textbf{I}_{r\times r} & | & \mathbf{0}_{r\times (m-r)} \\ ----- & | & ----- \\ \mathbf{0}_{(n-r)\times r} & | & \mathbf{0}_{(n-r)\times (m-r)} \\ \end{pmatrix}. $$

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Dim(Im(A))+Dim(Ker(A))=Dim(E)$\not\Rightarrow$E=Im(A)$\oplus$Ker(A) in general. See A(x,y)=(x-y,x-y) and take w=(1,1) then w$\in Ker(A)\cap Im(A)$ – Ivan3.14 Jun 25 '11 at 20:42
@missing314: No, $E/\ker A\cong\text{im} A\cong\widetilde{\text{im}}A:=E-\ker A$. I never stated $\text{im}A\oplus\ker A=E$, that could only be the case for endomorphism $A:E\rightarrow E$, but your target is other $F$. What I mean by $\widetilde{\text{im}}A$ is the complementary subspace of $\ker A$ within $E$ whereas for nonendomorphism you have $\text{im}A$ subspace of $F$, so the equation you write does not make sense. The rank-nullity th. is a result from the first isomorph. theorem since the dim of a quotient is $\dim (E/\ker A)=n-\dim\ker A=n-r$ which is precisely $\text{rank}A$. – Javier Álvarez Jun 25 '11 at 21:39
@missing314: the example you mention has $\ker A=<(1,1)>=\text{im} A$, but what I say is that $\widetilde{\text{im}}A=\mathbb{R}^2-<(1,1)>=<(-1,1)>$, i.e. the complementary of $\ker A$ satisfies $\widetilde{\text{im}}A\cong\text{im}A\cong E/\ker A\Rightarrow E=\widetilde{\text{im}}A\oplus\ker A$ which is still true for your example by the isomorphism $\widetilde{\text{im}}A\ni (-x,x)\mapsto (x,x)\in\ker A$. This is because of the special case when $\text{im}A\subseteq\ker A$ for endomorphisms, but for the GENERIC CASE $\text{im}A\cap\ker A=\{0\}$ and thus $\text{im}A=\widetilde{\text{im}}A$. – Javier Álvarez Jun 25 '11 at 22:10

Let $\left\{ u_1, \dots , u_n\right\}$ be a basis of $E$. Then, $\left\{ Au_1, \dots , Au_n \right\}$ is a system of generators of the subspace $\mathrm{im} A \subset F$. Whenever you have a system of generators of a vector (sub)space, you can delete some of them in order to obtain a basis. Since $r= \mathrm{rank} A = \mathrm{dim}\ \mathrm{im}A$, you can take $r$ of those $Au_i$ to form a basis of $\mathrm{im}A$. Reordering the original basis $u_1, \dots , u_n$ if necessary, we can assume that these are the first ones. So $Au_1, \dots , Au_r$ are a basis of $\mathrm{im}A$.

Now, you have $r\leq m$ linearly independent vectors $Au_1, \dots , Au_r$ in $F$. You can always complete a set of linearly independent vectors in order to order to obtain a basis of your vector space. So, choose $m-r$ vectors $v_{r+1}, \dots , v_m \in F$ such that $Au_1, \dots , Au_r, v_{r+1}, \dots , v_m$ is a basis of $F$.

And you're done: $\left\{ u_1, \dots , u_n\right\} \subset E$ and $\left\{ Au_1, \dots , Au_r, v_{r+1}, \dots v_m\right\} \subset F$ are bases you were looking for.

EDIT. I'm afraid my answer is wrong. If you perform the steps in it, the matrix you'll obtain looks like

$$ \begin{pmatrix} 1 & 0 & \dots & 0 & a^1_{r+1} & \dots & a^1_n \\ 0 & 1 & \dots & 0 & a^2_{r+1} & \dots & a^2_n \\ \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & 1 & a^r_{r+1} & \dots & a^r_n \\ 0 & 0 & \dots & 0 & 0 & \dots & 0 \\ \vdots & \vdots & \ddots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \dots & 0 & 0 & \dots & 0 \end{pmatrix} $$

But you have no control on the remaining $a^i_j$.

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The remaining $\{v_{r+1},...,v_m\}\subset F$ do not matter as long as you choose a basis of $E$ formed by the $u_1,...,u_r$ and choose the remaining $\dim\ker A=n-r$ vectors belonging to $\ker A$. In this way their image under $A$ is null in any basis of $F$ (in particular in the one constructed by completing $Au_1,...,Au_r$ with $v_{r+1},...,v_m$) and therefore your last $n-r$ columns of your matrix are all zeros whereas the first $r$ columns are as you obtained. – Javier Álvarez Jun 26 '11 at 18:17
Yes, but I should re-write all my answer. And you already did it. :-) – a.r. Jun 28 '11 at 23:19

This is a mixed of Javier and Agustí answers, I can take any basis of E, say $\{v_{1},\cdots v_{n}\}$ then look at the images $\{Av_{1},\cdots Av_{n}\}$ and find a basis of the image here with this basis $\{Av_{1},\cdots Av_{r}\}$ propertly ordered is obvious that the members of $\{v_{1},\cdots v_{r}\}$ don't belong to $Ker(A)$ then I can take a basis of the kernel as the dimension of it is DimE-Dim(Im(A))=n-r, let that basis be $\{v'_{r+1}\cdots v'_{n}\}$ then is easy to see that the basis asked for E is $\{v_{1},\cdots v_{r},v'_{r+1}\cdots v'_{n}\}$ and for F the basis would be $\{Av_{1},\cdots Av_{r},w_{r+1}\cdots w_{m}\}$ as I can always complete the basis of the image to a basis of the whole space F.

Thanks both (Javier and Agustí)

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looking well at my asnwer seems like there is a gap in the part where a I choose a basis of the kernel and joint it with the $\{v_{1}\cdots v_{r}\}$ as this doesn't guarantee me that the whole $\{v_{1}\cdots v_{r},v'_{r+1},\cdots v'_{n}\}$ would be linear indenpent. So to correct this I think a basis of the kernel should be chosen first then complete to a basis of the whole E – Ivan3.14 Jun 25 '11 at 22:35
indeed, find a basis of the image by expressing $A\cdot v$ as a linear combination of independent vectors of $F$. Then complete it to a basis of $F$. Now, solve the system $A\cdot u=0$ giving you the basis of $\ker A$; complete it to a basis of $E$. Then in those basis $A$ has the required matrix form. – Javier Álvarez Jun 26 '11 at 1:35

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