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100年 - 100 淡江大學 轉學考 線性代數#55896

科目：轉學考-線性代數 | 年份：100年 | 選擇題數：0 | 申論題數：14

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【已刪除】 (a) Find the matrix representation of T with respect to the standard basis {(1,0,0), (0,1,0), (0,0,1)} of 
【已刪除】(b) If (a, b，c) is a vector in ，what are the conditions on a, b, c that the vector is in the range of T? What is the rank of T?
(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
(a) Eigenvalues of T are either 0 or 1.
【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 
【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.