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13 Cards in this Set
- Front
- Back
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Rank Theorem |
If matrix A has n columns, then Rank A + dim Nul A =n |
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Basis Theorem |
Any linearly indep set of p element is a basis for H. Any set of p elements that spans H is a basis for H |
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Invertibility and Determinance |
Invertible iff det A != 0 |
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Invertible Matrix Theorem M-R |
cols of A form a basis of R^n Col A = R^n dim Col A = n rank A = n Nul A = {0} dim Nul A = 0 |
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Rank Definition |
Rank A = Dim Col A |
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Dimension Definition |
# of vectors in any basis |
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Basis of A |
Basis of A = pivot cols of A |
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Basis Definition |
Linearly independent set in a subspace that also spans that subspace |
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Null Space Definition |
Set of all solutions to Ax=0 |
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Column Space Definition |
Col A = set of all linear combinations of the cols of A |
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Subspace Requirements/Properties |
contains 0 vector for all U, V, U+V E H for all c, U, cU E H |
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Invertible Matrix Theorem A-F |
A is invertible A is row equivalent to In A has n pivot positions Ax=0 has only trivial solution Columns of A form a linearly independent set x->Ax is one to one |
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Invertible Matrix Theorem G-I |
Ax=b has atleast one solution for each b in R^n Cols of A span R^n x->Ax maps R^n onto R^n CA=I AD=I A^T is invertible |
