線性代數考古問題

Question

Let V≠{0} and W be both veator space of dimension n and m,

respectively.If T is a linear transformation from V onto W, show that the kernel of T is of dimension n-m

Update:

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Answer 1

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或許可以這麼證:

我先簡化問題

Let dim(V) = n and dim(W) = m. T:V→W is a linear transformation from

V onto W.

Claim: dim(N(T)) = n - m

Pf.

By Sylvester' theorem , we know dim(V) = dim(N(T)) + dim(R(T))

Since T is onto , so R(T) = W and R(T) is subspace of W , so

dim(R(T)) = dim(W) = m

Hence,

dim(N(T)) = dim(V) - dim(R(T)) = n - m #