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題組內容

4. (15 points) Let T be a linear operator on a finite dimensional vector space V. Suppose T is idempotent，that is T² = T. Prove that

(c) T is diagonalizable.

其他申論題

【已刪除】 (a) Show that for any is linearly dependent.

【已刪除】(b) Show that A is invertible if and only if I belongs to Span

(a) Eigenvalues of T are either 0 or 1.

【已刪除】(b) V = ker(T) range(T).

(a) Let T : V →W be a linear transformation from vector space V to vector space W. Show that T is nonsingular (1-1) if and only if T maps a linearly independent set of vectors in F to a linearly independent set of vectors in W.

(b) Let T : V→W be a, linear transformation from vector space V to vector space W. Suppose dimV = dimVK. Show that T is one to one if and only if T is onto.

【已刪除】6. (15 points) Let A = . Determine whether A similar to a diagonal matrix over If so, exhibit a basis for such that A is similar to a diagonal matrix.

【已刪除】7. (10 points) Let A be a, n x n matrix over the field be two distinct eigenvalues of A and W1,W2 be the corresponding eigenspaces for respectively. Show that

【已刪除】8. (10 points) Let V be a, finite dimensional vector space over a field F and dim V≥ 2. Let T : V→ 1/ be a linear transformation. If there exists a vector v G V such that V is spanned by prove that the characteristic polynomial of T is equal to it minimal polynomial.

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