Does Every Matrix with a Right Inverse Span Rm?

[Bob]

An m*n matrix A is said to have a right inverse if there exists an n*m matrix C such that AC=Im. A is said to have a left inverse if there exists an n*m matrix D such that DA=In.

(a) If A has a right inverse, show that the column vectors of A span Rm.
(b) Is it possible for an m*n matrix to have a right inverse if n<m? n>=m? Explain.

? How to do that? I dont' know how to start? what if A doesn't span Rm?

 

[nocturnal]

Note that the jth column of AC is equal to $$Ac_j$$ where [tex]c_j[/itex] is the jth column of C.

Think of the Linear transformation $$L_A:\mathbb{R}^n \rightarrow \mathbb{R}^m$$ defined as left multiplication by A. Another way to phrase the above statement is that the jth column of AC is equal to $$L_A(c_j)$$

Note that the columns of AC are linearly independent.

What does this tell you about rank(A)?

 

[Bob]

It's min{m,n}

thanks

 

[nocturnal]

It's min{m,n}

thanks

It is unclear to me based on your result whether you have actually arrived at a correct solution. To say that rank(A) = min{m,n}, while technically not wrong, leads me to suspect that you don't fully understand the problem, but before I make any accusations perhaps you could show me your work stating clearly your line of reasoning.

 

[Bob]

AC=Im

rank(A)=m

column vectors of A span Rm

if n>m, column vectors are linear dependent.
if n<m, AC does not equal Im, the reduced row echelon form of AC has at least one all zero row.

 

[nocturnal]

What you have written is correct, but you left out many of the details of the proof. In you're actual writeup you should fill in the gaps by either proving these statements or citing pre-existing theorems that allow you to make these jumps.

For example what are the justifications for going from AC = I am to rank(A) = m, and then how does this lead you to conclude that the column vectors of A span Rm?

Why if n<m does A not have a right inverse? What you wrote is true but is not an explanation.

aside:
If A has a right inverse, saying rank(A) = min{m,n} is the same as saying rank(A) = m. Do you see why?

regards,
nocturnal

 

[Bob]

If A has a right inverse

$$AC=(e_1 ,e_2 ,...,e_m)$$is consistent

$$Ac_1 = e_1$$
$$Ac_2 = e_2$$
...
$$Ac_m = e_m$$

an arbitrary vector in $$R_m$$ can be written as

$$\alpha _1 e_1 + \alpha _2 e_2 + ... + \alpha _m e_m$$

(Therom:A liner system Ax=b is consistent if and only if b is in the column space of A)

since $$e_1,e_2,...,e_m$$ are in the column space of A,
$$\alpha _1 e_1 + \alpha _2 e_2 + ... + \alpha _m e_m$$ is in the column space of A.

the column vectors of A span $$R_m$$