线代23秋期中试题答案 发布版

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Q1 (1)A (2)B（存疑, 一说D） (3)D (4)B (5)D

Q2(1)21

(2)

(3)1or-3 （存疑，一说 1 or -2）

(4)

(5)

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Q3 A basis for $$\left{\begin{bmatrix}1\0\-1\2\end{bmatrix},\begin{bmatrix}2\1\1\0\end{bmatrix},\begin{bmatrix}1\1\3\1\end{bmatrix}\right}.$$ A basis for

A basis for

A basis for

Q4 Gaussian Eliminations give:

If , then , has no solution.

If and has a unique solution.

If , has infinitely many solutions

Q5 (a)Let , than we have

(b)

Therefore,the matix representation of T with respect to, and , is:

(c)Since ,We can take X to be

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Q6 Apply Elementary Row and Column Operations to A and C to obtain for A and for C.

Where . Let . Then M can be converted to via elementary row and column operations.

Furthermore, the pivots in and can be used to eliminate the nonzero entries in , to obtain

In conclusion,