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Answer from [[User:Park296|Eun Young]] : | Answer from [[User:Park296|Eun Young]] : | ||
| − | No, it doesn't need to be reduced. If { [4 -2 6] } is a basis for some vector space <math>V</math>, then { [2 -1 3] } is also a basis for <math>V</math> and vice versa. A basis for a vector space is not unique but a dimension of a vector space is unique. | + | No, it doesn't need to be reduced. If { [4 -2 6] } is a basis for some vector space <math>V</math>, then { [2 -1 3] } is also a basis for <math>V</math> and vice versa. |
| + | If v belongs to span{[4 -2 6 ]}, v = c[4 -2 6] = 2c [2 -1 3 ] for some c. Hence, v belongs to span{[2 -1 3]}. | ||
| + | The opposite direction is same. So, span{[4 -2 6]}= span{[2 -1 3]}. | ||
| + | A basis for a vector space is not unique but a dimension of a vector space is unique. | ||
Revision as of 06:16, 29 August 2013
Homework 2 collaboration area
Here is the Homework 2 collaboration area. Since HWK 2 is due the Wednesday after Labor Day, I won't have a chance to answer questions on Monday like usual. I will answer any and all questions here on the Rhea on Tuesday with help from Eun Young Park. - Steve Bell
Questions from a student :
When finding a basis, does it always have to be fully reduced? For example, if you have a basis [4 -2 6] does it need to be reduced to [2 -1 3] or is either answer acceptable? Jones947
Answer from Eun Young :
No, it doesn't need to be reduced. If { [4 -2 6] } is a basis for some vector space $ V $, then { [2 -1 3] } is also a basis for $ V $ and vice versa. If v belongs to span{[4 -2 6 ]}, v = c[4 -2 6] = 2c [2 -1 3 ] for some c. Hence, v belongs to span{[2 -1 3]}. The opposite direction is same. So, span{[4 -2 6]}= span{[2 -1 3]}. A basis for a vector space is not unique but a dimension of a vector space is unique.
