Vector Spaces Definition (wiki)

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練習題

Explain why the following sets of three-by-one matrices (with real number scalars) are vector spaces:

  1. The set of three-by-one matrices with zero in the first row;
  2. The set of three-by-one matrices with first row equal to the second row;
  3. The set of three-by-one matrices with first row a constant multiple of the third row.

Sol: 根據定義要確認 vector space 的 8 個條件, 其中只需確定 $\mathbf{0}$ 和 $-\mathbf{v}$ is closed 即可, 其他性質很明顯成立

Linear Independence

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從定義也可以知道, if $c_1\mathbf{u}_1+c_2\mathbf{u}_2+...+c_n\mathbf{u}_n=\mathbf{0}$, 存在一解 $c_1\neq0,c_2=c_3=...c_n=0$, 則 $u_1=\mathbf{0}$.

得到 $\mathbf{0}$ 跟任何 vectors 都是 linear dependent.

可以利用 rref (reduce row echelon form) 來確認矩陣可逆性判斷是否線性獨立

Span, Basis and Dimension

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練習題

Which one of the following is an orthonormal basis for the vector space of all three-by-one matrices with the sum of all rows equal to zero?

Gram-Schmidt process