[理工] 線代 關於空間的"包含於"
Let A be an m x n matrix and B an n x k matrix.
Show that the column space of AB is a subspace of the column space of A.
解答:
對於每一個 y 屬於 CS(AB), 存在x屬於R^k使得y=(AB)x
→ y=A(Bx)屬於CS(A)
所以CS(AB) 包含於 CS(A)
因為CS(AB)及CS(A)皆為R^m的子空間，所以CS(AB)為CS(A)的子空間。
我想問的是"y=A(Bx)屬於CS(A),所以CS(AB) 包含於 CS(A)"
為什麼不是CS(A)包含於CS(AB)呢?
另外也是類似的問題
證明rank(AB)<=rank(A)
T T T
解法是: A*x=0 => B*A*x=0
T T T
所以N(A )⊆N(B A ) 不太懂包含於的大小關係@@
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