A question in complementing into other basis

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In summary, the conversation discusses adding independent vectors to extend the basis of a smaller space to a basis for a larger space. The concept of orthogonal complement is mentioned but dismissed, and the answer in the book is to simply add the missing vector to make the bases equal. An example is given of adding (1,2,0,1) to the basis in order to achieve this.
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  • #2
I'm not sure what you mean by "complement" the base. I suspect you mean just to "extend" the basis of the smaller space to a basis for the larger. If that is the case you would just add independent vectors that are still the larger space. But that seems awfully trivial! You have, correctly, that a basis for [itex]U\cupV[/itex] is {(1, -1, 0, 0)} and you were told that a basis for U is {(1, 2, 0, 1), (1, -1, 0, 0}. You need to add a vector to {(1, -1, 0, 0} to give a basis for U? Duh!

Extending the basis of U to a basis of [itex]U\cup V[/itex] is similarly trivial. LOOK at the bases you already have!

I am wondering if you are not talking about orthogonal complement. That would mean you need to add a vector, in the larger space, that is orthogonal (perpendicular) to the vectors already in the basis.
 
  • #3
no its not orthogonal complement
the answer in the book is that they just add the missing vector
(the vectors which lack the small groop)
and makes the equal

for example in the example that you presented they add (1,2,0,1)
is that ok?
 

Related to A question in complementing into other basis

1. What is "complementing into other basis"?

"Complementing into other basis" refers to a mathematical technique where a given vector space is represented in terms of a different basis set. This is used to simplify calculations and provide a different perspective on the data.

2. Why is "complementing into other basis" important?

This technique is important because it allows for easier analysis and manipulation of data. By changing the basis set, certain properties of the data may become more apparent, making it easier to solve problems and make predictions.

3. How is "complementing into other basis" different from standard vector space calculations?

The main difference is that "complementing into other basis" involves transforming the basis set, while standard vector space calculations use the original basis set. This transformation can make calculations simpler and provide a different perspective on the data.

4. What are some common applications of "complementing into other basis"?

This technique is commonly used in fields such as linear algebra, quantum mechanics, and signal processing. It can also be applied in data analysis, machine learning, and computer graphics.

5. Are there any limitations to "complementing into other basis"?

Like any mathematical technique, there are limitations to "complementing into other basis". It may not always be possible to find a suitable basis set for a given vector space, and the calculations can become complex for higher dimensional data. Additionally, the results obtained may not always be intuitive or easy to interpret.

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