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研究所、轉學考(插大)◆線性代數
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110年 - 110東吳大學_碩士班招生考試_數學系︰線性代數#100057
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1. (a) In R
3
, represent all the linear combinations of v=(2,0,1) and w=(0,1,2), give a geometric interpretation. And find a vector which is not a combination of v and w.
其他申論題
9. (8 points) Evaluate
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10. (8 points)Find f(x) =
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11. (10 points) Let f'(x) exist everywhere. If f(2) = 8 and f'(x) ≤ 5 for all x. Use the mean value theorem to answer how small can f(0) possibly be?
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12. (10 points) Let , x≥ 0. Use the mean value theorem for definite integral and the limit-definition of derivative to find f'(x). (Mean Value Theorem) Suppose f : [a,b] → IR is continuous on fa, b and differentiable on (a,b). Then there exists c (a,b) such that(Mean Value Theorem for Definite Integral) Suppose f : [a,b] → IR is a continuous function. Then there exists c [a,b] such that
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(b) In the following two planes, which one is a subspace of R3 (give your reasons) and find a basis for the subspace. x +2y-3z -4=0, 2x -4y +5z =0
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2. (a) If A is a 3 × 3 matrix has the determinant det(A) = -5, then what is det and det ?
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(b) Let E be an n × n elementary matrix, what is the determinant det(E) of E ?
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(c) 8% For the matrix , find the value of , where is the cofactor of entry
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3. For , find the reduced row echelon form of A and verify the dimension theorem for matrices.
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4. Find the eigenvalues and eigenvectors of the matrix . Is A diagonalizable? And find the Jordan form of A.
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