4. (20%) Let A be any m × n matrix. Prove that A^T A and A have the same null space. (Hint: Let v be a vector in R^n such that A^T A v = 0. Observe that A^T A v · v = A v · A v = 0)

1. (2% for each blank, 12% in total, 本題計算過程不計分) 　Suppose two infinite planes of charge are parallel to each other (Fig. 1), one positively charged and the other negatively charged. Both planes have the same charge density σ. The direction and the magnitude of electric field at point P1 (to the left of the positively charged plane) are (1a) and (1b) , respectively. The direction and the magnitude of electric field at point P3 (to the right of the negatively charged plane) are (1c) and (1d) , respectively. The direction and the magnitude of electric field at point P2 (between the two planes) are (1e) and (1f) , respectively.

4. (20%) Let A be any \( m \times n \) matrix. Prove that \( A^T A \) and A have the same null space. (Hint: Let \( \vec{v} \) be a vector in \( \mathbb{R}^n \) such that \( A^T A \vec{v} = 0 \). Observe that \( A^T A \vec{v} \cdot \vec{v} = A \vec{v} \cdot A \vec{v} = 0 \))

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