Calculating Subspace Dimension in R4: U = [a b c d] | a+b=c+d

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The discussion focuses on calculating the dimension and finding a basis for the subspace U in R4 defined by the equation a + b = c + d. The user proposes a basis consisting of the vectors (1,0,0,1), (0,1,0,1), and (0,0,1,-1), while the reference text provides a different basis: (-1,1,0,0), (1,0,1,0), and (-1,0,0,1). Both bases are valid, demonstrating that multiple bases can exist for the same subspace, which is a fundamental concept in linear algebra.

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Find a basis and calculate the dimension of the following subspaces of R4
U = [a b c d] | a+b=c+d in R

now a basis is supposed to be linear independent
but that conditon on a,b,c,d makes this a bit strange

can i just say d = a+b - c and then pick
a basis as (1,0,0,1),(0,1,0,1),(0,0,1,-1)

text says this however
(-1,1,0,0),(1,0,1,0),(-1,0,0,1)

please help
i am currently studying for my exam i would like to understand everything!

your help is greatly appreciated!
 
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It looks like your text just chose to write a = d + c - b. I'm sure you know there can be more than one basis.
 

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