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The opposite direction is same. So, span{[4 -2 6]}= 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. | A basis for a vector space is not unique but a dimension of a vector space is unique. | ||
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| + | Question from a student: | ||
| + | |||
| + | From what I understand, a basis is a set of vectors that can be used to | ||
| + | create any vector in the span. So for example, if the basis is [1 0] [0 1], | ||
| + | then the span could be [1 0] [0 1] [2 2] [2 0]. Is that correct? | ||
| + | |||
| + | Answer from [[User:Bell|Steve Bell]]: | ||
| + | |||
| + | The span is ALL vectors you get by taking linear combinations. | ||
| + | Hence, the span is x*[1,0] + y*[0,1] = [x,y] as x and y range | ||
| + | over all possible values, i.e., the span is R^2. | ||
Revision as of 06:23, 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.
Question from a student:
From what I understand, a basis is a set of vectors that can be used to create any vector in the span. So for example, if the basis is [1 0] [0 1], then the span could be [1 0] [0 1] [2 2] [2 0]. Is that correct?
Answer from Steve Bell:
The span is ALL vectors you get by taking linear combinations. Hence, the span is x*[1,0] + y*[0,1] = [x,y] as x and y range over all possible values, i.e., the span is R^2.
