Which of the following sets of vectors spans?   1. Which of the following sets of vectors spans`r^2`?  a)      {(1,1), (-2,-2)} b)      {(1,1), (1,2)} c)       {(1,2), (1/2,2)} d)      {(-1,1), (1,-1)}   2. Which of the following sets of vectors spans `r^3`?  a)      {(1,1,1), (2,2,2)} b)      {(1,3,1), (2,2,2)} c)       {(1,2,1), (1/2,1,1/2)} d)      {(1,3,2), (-1,-3,-2)}

Expert Answers

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You asked several questions. I will just answer some of them. Use the same method to solve the rest.

 

2 vectors spans `RR^2` iff they are linearly independant iff their determinant is not 0.

 

Let's find their determinant.

For 2 vectors (a,c) and (b, d) their determinant is

`det([[a,b],[c,d]])=ad-bc`

a)`det([[1,-2],[1,-2])=(-2)-(-2)=0`

They are linearly dependant, they don't span `RR^2`

 

b) `det([1,1],[1,2]])=2-1=1ne0`

They are linearly independant, they do span `RR^2`

 

Find the determinant of the other vectors using the formula, compare the result to 0 and find that  the vectors of c) span `RR^2`

but not the vectors of d).

The sets b) and c) span `RR^2`

 

 

Second question. To span `RR^3` , you need at least 3 vectors. You have only sets of 2 vectors therefore no sets of 2 vectors will span `RR^3`

 

 

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