→ njru81l :你應該看一下"一組"向量線性相依的定義 04/24 12:49
→ njru81l :給a,b 是什麼用的，c就是什麼用的 04/24 15:19
1.
Let S = {v1, v2, … , vn} be a set of vectors in a vector space V.
Let w be a vector in V such that
w = c1v1 + c2v2 + … + cnvn, where ci are scalars and c1 ≠0.
Prove that T = S U {w} is a linearly dependent set.
這題我是用
w=c1v1+ c2v2 +...+ cnvn
WLOG assume
v1=c2v2+....+cnvn
V1-c2v2-....-cnvn=0
since c1=1, this is non-teivial linearly combinations
so w is linearly dependent
但我這裡就不知道怎麼證 SＵ{w}是linearly dependent set..
懇請板上神人指點，我這邊觀念頗弱的
2.
Consider the vector space M3,3 with the usual operations. Determine whether
each of the following are proper subspaces of this vector space. If yes,
provide a proof, if no explain why.
a. The set of all matrices where a11 = 0.
b. The set of all invertible matrices.
c. The set of all matrices where the sum of every row and column is the same
constant.
Examples for elements of the set in c –
The zero matrix, every row and column sums to 0.
The identity matrix, every row and column sum to 1.
The matrix
2 0 2
0 4 0
2 0 2
- every row and column sums to 4. There are many other
Examples. All of these are member of the set defined in c.
其實我不太懂這題的意思，也不知道他下面給的條件(c)是要幹嘛
先謝謝板上神人賜教
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