Prove Rk(A+B) ≤ Rk(A) + Rk(B) - Tal

[talolard]

Hey Guys, Another matrice question

Homework Statement

Prove: Rk(A+B)$$\leq$$ Rk(A) +Rk(B)

The Attempt at a Solution

Rk(A+B) = Dim[R(A) + R(B)]
Where R(A) is the row space of A
we know that Dim[R(A)+R(B)] = Dim[R(A)] + Dim[R(B)] - Dim[R(A)$$\cap$$R(B)]
Which means that Dim[R(A)+R(B)] $$\leq$$ Dim[R(A)] + Dim[R(B)] iff Rk(A+B)$$\leq$$ Rk(A) +Rk(B)

I heard a rumor that this can also be done with linear transformations, can anyone elighten me on that path?

Is this correct?
Thanks
Tal

 

[HallsofIvy]

Hey Guys, Another matrice question

Homework Statement

Prove: Rk(A+B)$$\leq$$ Rk(A) +Rk(B)

The Attempt at a Solution

Rk(A+B) = Dim[R(A) + R(B)]
Where R(A) is the row space of A
we know that Dim[R(A)+R(B)] = Dim[R(A)] + Dim[R(B)] - Dim[R(A)$$\cap$$R(B)]
Which means that Dim[R(A)+R(B)] $$\leq$$ Dim[R(A)] + Dim[R(B)] iff Rk(A+B)$$\leq$$ Rk(A) +Rk(B)

I heard a rumor that this can also be done with linear transformations, can anyone elighten me on that path?

If F is a linear transformation from U to V, then, given specific bases for U and V, there exist a matrix representing F so essentially we can interpret matrices as being linear transformations and vice versa. Any thing true of matrices is true of linear transformations.

Is this correct?
Thanks
Tal