[線代] orthogonal projection

[mqazz1 (無法顯示)]

1. If w is the orthogonal projection of a vector v in R^n onto a subspace W of

R^n then w is orthogonal to v

        False

========

2. Let W be a subspace of R^n and v be a vector in R^n. Among all vectors in W,

                                                               +
the vector closet to v is the orthogonal projection of v onto W

        False

=========

3. The set of all vectors in R^n orthogonal to one fixed vector is a subspace

of R^n

        True

================
                                          +
4. If W is a subspace of R^n, then W and W  have no vectors in common

        False

=========

5. If a square matrix has orthonormal columns, then it also has orthonormal rows

        True


請問這些是為甚麼呢?

謝謝

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◆ From: 218.166.113.145
推 jacky7987   :R^3 P(x,y,z)=(x,y,0) is a projection
→ jacky7987   :(x,y,z)‧(x,y,0)≠0
→ jacky7987   :3. 就照定義證明Given u. Suppose W={v|v垂直u}
→ jacky7987   :(av_1+bv_2)‧u=0
→ jacky7987   :and 0屬於W
→ jacky7987   :4 0 in W and W┴
→ jacky7987   :A^TA=I A^{-1}=A^T hence AA^T=I
→ jacky7987   :上面是第五題
→ jacky7987   :第二題最靠近的好像是project to W