# Span question with Restrictions

• jberg074
In summary, the conditions for [a b c d] (transposed) to be in the span of the given vectors are a=a, b=b, c=2b-a/2, and d=4b+a/2, where a, b, c, and d are real numbers.
jberg074

## Homework Statement

Let U be the span of
[ 0 ] [ 2 ]
[ 1 ] and [ 0 ]
[ 2 ] [ -1]
[ 4 ] [ 1 ]

Give conditions on a, b, c and d so that [a b c d] (transposed) is in U (as in, give restrictions that are equations of only a, b, c and d.

## The Attempt at a Solution

I've set up the problem to set the two given vectors multiplied by scalar x and y, yielding four equations:
2y = a,
x = b,
2x - y = c, and
4x + y = d.

Is this the right first step to take? After that, I take x or b as a free variable, and try to rewrite the other equations in terms of b, yielding:

2y = a, b = b, 2b - y = c, and 4b + y = d.

After this, however, I am stuck. Any pointers? Did I set up the problem the right way?

Edit: I also know that [a b c d] must be a linear combination of the two given vectors if it is in the span. However, if it is a linearly independent set, the trivial zero vector could be matched with [a b c d], although I am pretty sure the answer isn't that simple.

Last edited:
Hi jberg074!

You also know that 2y=a, so $y=\frac{a}{2}$. Substitute that into the equations

2b - y = c
4b + y = d.

and you'll find your conditions on a,b,c and d.

Is the answer as simple as this?
a=6b-c
b=b
c=2b-a/2
d=4b+a/2

jberg074 said:
Is the answer as simple as this?
a=6b-c
b=b
c=2b-a/2
d=4b+a/2

I'm not sure where you got the a=6b-c, actually. But the two last equations are what you need. These give necessary and sufficient conditions for something to be in the span. In general, the span is given by

$$\{(a,b,2b-a/2,4b+a/2)~\vert~a,b\in \mathbb{R}\}$$

$$\{(a,b,2b-a/2,4b+a/2)~\vert~a,b\in \mathbb{R}\}$$[/QUOTE]

I used other questions to eliminate variables and to try and get a in terms of the other. However, in the text above, shouldn't "a" be replaced by "a/2"?

jberg074 said:
$$\{(a,b,2b-a/2,4b+a/2)~\vert~a,b\in \mathbb{R}\}$$

I used other questions to eliminate variables and to try and get a in terms of the other. However, in the text above, shouldn't "a" be replaced by "a/2"?[/QUOTE]

No, we have concluded that

a=a
b=b
c=2b-a/2
d=4b+a/2

So, we will have

$$(a,b,c,d)=(a,b,2b-a/2,4b+a/2)$$

Ah, I understand now. I wasn't exactly sure what the question was asking for at first; thank you very much for the help!

## 1. Can you explain what a "Span question with Restrictions" is?

A "Span question with Restrictions" is a type of research question in which the scope or parameters of the question are limited or restricted in some way. This allows for a more focused and specific inquiry, often leading to more precise and meaningful results.

## 2. What are some examples of restrictions that can be applied to a Span question?

Restrictions can include limitations on the time frame, geographic location, population, or variables being studied. For example, a Span question with the restriction of time frame may only focus on data from the past 5 years, while a restriction on population may only consider data from a specific demographic.

## 3. How do Span questions with Restrictions differ from other types of research questions?

Unlike open-ended questions, Span questions with Restrictions have a more defined and specific focus. They also differ from closed-ended questions in that they still allow for some flexibility and variation within the restrictions set by the researcher.

## 4. What are some benefits of using Span questions with Restrictions in research?

One major benefit is the ability to control for extraneous variables and potential confounding factors, leading to more reliable and valid results. Span questions with Restrictions also allow for more efficient use of resources, as researchers can focus on collecting and analyzing data that is most relevant to their inquiry.

## 5. Are there any potential drawbacks or limitations to using Span questions with Restrictions?

One potential drawback is that the restrictions may limit the generalizability of the results. Additionally, if the restrictions are too narrow or specific, it may be difficult to find enough data or participants to adequately address the research question. It is important for researchers to carefully consider the restrictions and their potential impact on the study.

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