所屬科目：【阿摩】未分類題庫
1. (10%) Use Gauss-Jordan reduction to solve the following system:$$\left\{ \begin{array}{l} x _ {1} - 2 x _ {2} - x _ {3} + 2 x _ {4} = 0 \\ 2 x _ {1} - x _ {2} + 3 x _ {3} + x _ {4} = 9 \\ - x _ {1} + x _ {2} - x _ {3} - x _ {4} = - 4 \end{array} \right.$$
(a)(10%) Find a basis for the row space of \( \mathbf{A} \) and a basis for the null space of \( \mathbf{A} \).
(b) \( (5\%) \) Verify the rank-nullity theorem for A.
3. (10%) Let \( \mathbf{S} \) be the two-dimensional subspace of \( \mathbf{R}^3 \) spanned by \( \mathbf{x}1 = [1, 0, 2]^{\mathrm{T}} \) and \( \mathbf{x}2 = [0, 1, -4]^{\mathrm{T}} \). Find a basis for \( \mathbf{S}^\perp \).
4. (15%) Let \( \mathbf{B} \) be an \( m \times n \) matrix, and the dimension of the row space of \( \mathbf{B} \) is \( n \). Show that the matrix \( \mathbf{B}^{\mathrm{T}}\mathbf{B} \) is symmetric positive definite.
5. (10%) Determine whether the following argument is valid or not valid with a proof.Rainy days make gardens grow.Gardens don't grow if it is not hot.It always rains on a day that is not hot.Therefore, if it is not hot, then it is hot.
6. (10%) A pair of dice, each with the numbers 1, 3, 5, 7, 9, 11, on its six sides are rolled. What is the expected value of the sum of the numbers showing on this pair of dice? What is the expected value of the product of the numbers showing on this pair of dice?
7. (20%) Consider the following graph. Is it a planar graph? Does it have a Euler circuit? Does it have a Euler path? Does it have a Hamilton circuit? Does it have a Hamilton path? Answer these questions and prove your answers.
8. (10%) Consider the following recursive function XXX where the global variable count is initialized to 0 and input n is a positive integer. What is the final value of count after XXX(5) is executed? Furthermore, in general, what is the final value of count as a function of n?XXX(n){if (n = 1) or (n = 2) thencount ← count + 1else{XXX(n-2)XXX(n-1)XXX(n-2)count ← count + 1}}
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