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Finding basis of spaces and dimension 
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#1
Feb2811, 08:50 PM

P: 27

1. The problem statement, all variables and given/known data
Find a basis for each of the spaces and determine its dimension: The space of all matrices A=[a b, c d] (2x2 matrix) in R^(2x2) such that a+d=0 2. Relevant equations 3. The attempt at a solution So I jumped at this question without knowing too much about spaces and dimensions, but: I think a possible combination of basis can be: [1 0, 0 0]. [0 1, 0 0]. [0 0, 1 0] (not sure if [0 0, 0 1] would be considered since d would be negative in this case) Also from that I assume the dimension is 3? Truthfully even if I got it right, I do not even know what happened. Can someone conceptually tell me what I am doing exactly? Thanks in advance. 


#2
Feb2811, 08:54 PM

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P: 3,307

you need to come up with a basis, that has elements, such that any 2x2 matrix with a=d can be written as a linear sum of the basis elements.
clearly the basis elements will need to satisfy being a 2x2 matrix with d=a, you first element does not 


#3
Feb2811, 08:55 PM

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P: 3,307

note this is entirely equivalent to considering 4 component vectors in R^4, with x_1 = x_4
often the vector form is easier to conceptualise 


#4
Feb2811, 09:04 PM

P: 27

Finding basis of spaces and dimension
Also since I am actually having a hard time learning this on my own, can you tell me what you mean by "elements" and "linear sum"? 


#5
Mar111, 01:52 AM

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P: 3,307

if you're not familair with those terms, you may need to do a bit of reading.. though i have been a little loose with terminiology in post #4 i actually meant component and have changed accordingly a basis, is a set of vectors that spans a vector space an element is a member of a set, for example a vector in the basis set a linear combination (or sum) is a vector addition with scalar multiplication eg. if u,v are vectors, and a,b are scalars, then w = au + bv is a linear combination a set, S, of vectors spans a space if any vector in the space can be written as a linear combination of vectors in S 


#6
Mar111, 07:03 AM

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P: 39,344

From a+ d= 0 you get d=a as you say. You can then write
[tex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}= \begin{bmatrix}a & b \\ c & a\end{bmatrix}[/tex] [tex]= \begin{bmatrix}a & 0 \\ 0 & a\end{bmatrix}+ \begin{bmatrix}0 & b \\ 0 & 0\end{bmatrix}+ \begin{bmatrix}0 & 0 \\ c & 0\end{bmatrix}[/tex] and the dimension and a basis should be clear. 


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