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3. Let V and W be finite-dimensional vector spaces over field F with dim(V) = n...

3. Let V and W be finite-dimensional vector spaces over field F with dim(V) = n and dim(W) = m, and

let φ : V → W be a linear transformation. Fill in the six blanks to give bounds on the sizes of the

dimension of ker(φ) and the dimension of im(φ).

3. Let V and W be finite-dimensional vector spaces over field F with dim(V ) = n and dim(W) = m, and

let φ : V → W be a linear transformation. Fill in the six blanks to give bounds on the sizes of the

dimension of ker(φ) and the dimension of im(φ).

max(____,_____) ≤ dim(ker(φ)) ≤_____ _____≤ dim(im(φ)) ≤ min(_____,_____)

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Answer #1

. and . Then,

Of course, . and also . Because the subspce is always subset of . Also, can send at max whole space to . So .

, because can be the map. Also, . Image can not exceed the space .

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