Linear Algebra (linear transformation)

👨🏻‍🏫 學術台數學

pollyconstant

2008-08-09 7:54 pm

Let L: V--> W. And S={v1, v2, .... vn} is a linearly independent set in V. If the set={L(v1), L(v2),.........L(vn)} is linearly independent in W. Proof that L is one-to-one.

更新1:

Modified the question as below: Let L: V-->W. If any linearly independt set in V map to W is also a linearly independent set, L is 1-1

更新2:

Modified the question as below: Let L: V-->W. If any linearly independt set in V map to W is also a linearly independent set, Proof that L is 1-1

回答 (2)

TUN SHUK

2008-08-12 2:20 am

✔ 最佳答案

Since S is a linearly independent set,
c1v1 + c2v2 +.... cnvn = 0
L{0} = L {c1v1 + c2v2 .... cnvn} = c1 L{v1} + c2 L {v2} ....cn L {vn}

Because S is a linearly independent set, c1=c2=...cn=0.
So, c1 L {v1} + c2 L {v2} .... =0

Therefore L{0} = 0 in W. There is a theorem states that if the zero vector in V is mapped to W and comes up a zero vector in W, L is one-to-one.

2008-08-12 12:11:08 補充：
The title is " LINEAR TRANSFORMATION"

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Andrew

2008-08-13 6:27 am

好無奈，title都算，唉～
下次睇問題要再小心d先得。

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