<br>由题设有方程组<br><img src="/s/tiw/p3/UpLoadImage/2013-05-07/8e68e7c1-759a-46b6-8dce-a7fcb912abf7.jpg" width="217" height="94"><br>该方程组的系数行列式为<br><img src="/s/tiw/p3/UpLoadImage/2013-05-07/321a81c1-c77f-47a0-bb5a-6f8b6dfd6ab2.jpg" width="410" height="96"><br>由克莱姆法则知方程组有解,所以α<sub>1</sub>,α<sub>2</sub>,…,α<sub>r</sub>可由β<sub>1</sub>,β<sub>2</sub>,…,β<sub>r</sub>线性表示,从而α<sub>1</sub>,α<sub>2</sub>,…,α<sub>r</sub>和β<sub>1</sub>,β<sub>2</sub>,…,β<sub>r</sub>等价,故有两向量组的秩相等.
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