Answer to Question #214227 in Linear Algebra for SID

Question

Let V and W be two vector spaces over the field F and T, T1, T2 be linear transformations from

V to W. Prove the following: (2+3=5 marks)

(a) rank(αT) = rank(T), for all α ∈ F and α 6= 0.

(b) |rank(T1) − rank(T2)| ≤ rank(T1 + T2) ≤ rank(T1) + rank(T2).

Answer 1

Let V and W be vector spaces. A linear transformation T : V → W is a function from V to W such that T(u + v) = T(u) + T(v) and T(cu) = cT(u) for all u, v ∈ V and for all c ∈ R

a)

rank(T) is dimension of T(v).

$\alpha T(v)=T(\alpha v)$

dimension of $T(\alpha v)=$ dimension of $T( v)$

So, rank(αT) = rank(T)

b)

since image of (T1+T2) is subspace of (image of T1+image of T2):

$rank(T1 + T2) ≤ rank(T1) + rank(T2)$

since image of T1 and image of T2 are subspaces of image of (T1+T2):

$|rank(T1) − rank(T2)| ≤ rank(T1 + T2)$

so,

$|rank(T1) − rank(T2)| ≤ rank(T1 + T2) ≤ rank(T1) + rank(T2)$