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17 Cards in this Set
- Front
- Back
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how to tell if one-to-one |
augment matrix with 0 has only trivial solutions |
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how to tell if onto |
augment matrix with b1,b2,... circle 1 in every column |
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how to find adj(A) |
1. find A' by matrix created in each part 2. cofactor(C) by chessboard signs 3. C(trans)=adj(A) rows become columns |
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how to find inverse(A) from adj(A) |
1/det(A) * adj(A) |
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Cramer's Rule to solve for linear equations |
1. Find delta = (det (A)) 2. Find delta(x1) = det(A) with answers in x1 3. Find delta(x2) = det(A) with answers in x2 4. Find delta(x3) = det(A) with answers in x3 5. x1 = delta(x1)/delta...etc |
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Consider the subset v1, v2, v3, .... Find a basis |
1. augment vectors with 0 2. dependent columns match vectors |
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how to write a product of elementary matrices |
1. augment with identity 2. track changes 3. write in reverse order (last to first) 4. diag = reciprocal, off-diag = opposite sign |
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Characteristic polynomial |
1. lambda-a -b -c lambda-d 2. find determinant |
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how to find eigen values from characteristic polynomial |
roots of polynomial, solve for zero
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how to find eigen vectors for each eigen value |
plug in each eigen value for lambda in determinant and augment with 0 |
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how to find change of coordinate matrix, P |
put vectors in B down in matrix |
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how to use P to find coordinate vector [v]b for v=[ ] |
1. Find P inverse 2. Multiply P inverse times vector given |
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how to find D such that P(inv)AP=D |
1. Find characteristic polynomial 2. Find roots 3. Plug in roots for lambda and augment with 0 4. Vectors = P 5. Find P inverse 6. P(inv)AP = D where A is original matrix |
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how to find [T]b relative to basis when T(x,y)=... |
1. Let A = vectors from T(x,y) 2. P = vectors from basis 3. Find P inverse 4. P(inv)AP = D = [T]b |
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how to find the basis of the nullspace of a matrix |
1. augment with 0 2. find solutions 3. write in vector form |
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how to find the basis of the column space of a matrix |
1. augment with 0 2. columns with circle 1s match vectors in original matrix |
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how to find the area of a triangle |
1. (k,m,n) (r,s,t) (u,v,w) 2. | 1 1 1 | |r-k s-m t-n| * (1/2) |u-k v-m w-n| |
