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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

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

其他申論題

(c) What are the conditions on a, b, c that the vector (a, b, c) is in the null space of T? What is the nullity of T?

【已刪除】2. (10 points) Let u = (2,1,0)，v = (3,0,2) and w = (0, -2,3). Suppose that T is a linear operator on that interchanges u and v, and maps w to (1,0,0). Find the matrix representation [T]B of T with respect to the standard basis B = {(1，0,0)，(0，1，0)，(0，0，1)}.

【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

【已刪除】(b) V = ker(T) range(T).

(c) T is diagonalizable.

(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

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