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war485
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Homework Statement
Just like to verify these statements as being always true or false since I've been told subspaces is the most important concept in matrix/linear algebra.
a. set of vectors in R3 that satisfy x = |y| form a vector space of R3
b. if S is a subspace of Rn and the dimension of S = n, then S = Rn
c. dimensions of Col A and Nul A add up to the number of columns of A
d. if a set of p vectors spans an x-dimensional subspace C of Rn, then these vectors form a basis for C
e. The dimension of Nul A is the number of variables in the equation Ax = 0
f. if C is an x-dimensional subspace of Rn, then a linearly independent set of x vectors in C is a basis for C
The Attempt at a Solution
a. this is not always true because the vectors might not contain the zero vector (not going through the origin), which is required for a vector subspace.
b. always true since S defines the basis so dimension of each vector in subspace is n
c. false, I know that Rank of matrix + nulity A = # columns of A
so number of columns = dimension of col(A) - dimension of nulity of A
d. think this is false, because span is linearly dependent, but a basis needs to have linearly independent vectors.
e. I think this one is always true by definition.
f. This one seems tricky, so I looked up some definitions. Since there's a subspace, I can pick out any linearly independent vectors from that and so it should always be a basis for C. So i think this is true.
I desperately want to understand this stuff fully with the right reasons.
Any help is greatly appreciated.