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29 Cards in this Set
- Front
- Back
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consistent inconsistent |
system has solutions system has no solutions |
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trivial solution |
solution to all homogeneous matrices Ax=0 |
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m x n matrix |
m rows, n columns (i corresponding to rows, j corresponding to columns) |
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zero matrix |
all elements equal 0 |
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identity matrix |
all elements equal 0 except those along the diagonal, which equal 1 |
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matrix addition |
A + B = C |
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scalar multiplication |
C = cA |
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subtraction |
A - B = A + (-1)B |
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summation notation |
sum[(a + b)c] = sum[ac] + sum[bc] sum[ca] = csum[a] sum[sum[a]] = sum[sum[a]] |
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linear combination |
sum[cA] = c1A1 + c2A2 + ... + ckAk |
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transpose |
swapping rows for columns |
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matrix multiplication |
AB = C where C = c where c = sum[aikbkj] |
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trace |
the summation of the matrix's diagonal |
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addition properties |
A + B = B + A A + (B + C) = (A + B) + C unique matrix 0 such that A + 0 = A unique matrix D such that A + D = 0 |
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multiplication properties |
(AB)C = A(BC) (A + B)C = AC + BC C(A + B) = CA + CB |
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scalar multiplication |
r(sA) = (rs)A (r + s)A = rA + sA r(A + B) = rA + rB r(AB) = (rA)B |
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transposition |
(A^T)^T = A (A + B)^T = A^T + B^T (rA)^T = r(A^T) (AB)^T = (B^T)(A^T) |
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diagonal matrix |
all elements equal 0, except those along the matrix's diagonal |
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scalar matrix |
if A is diagonal and if every element along the matrix's diagonal equals one thing (e.g., the identity matrix) |
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exponential properties |
A^0 = identity matrix A^P is defined only if A is a square matrix (A^P)^Q = A^(PQ) (AB)^P = (A^P)(B^P) only if AB = BA |
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upper triangular lower triangular |
if aij = 0 for i > j if aij = 0 for i < j |
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symmetric skew symmetric |
if A^T = A if A^T = -A |
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nonsingular/invertible |
A is a square matrix; A is nonsingular if there is a unique matrix A^(-1) such that A(A^(-1)) = (A^(-1))A = the identity matrix
A is singular if it is row equivalent to a matrix with a row of zeros |
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properties of inverse matrices |
(A^(-1))^(-1) = A (AB)^(-1) = (B^(-1))(A^(-1)) (A^T)^(-1) = (A^(-1))^T |
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the image of f |
f(u) = Au is called the image of f/A, where A is a matrix, f is a relation from one real set to another, and u is some column vector. Au will also be a column vector.` |
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row reduced echelon form |
1. all zero rows at bottom of matrix 2. first non-zero entry from left is a 1 3. leading ones appear to the right of and below leading ones in previous rows 4. all other entries in columns with leading ones are zeros |
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row equivalence |
B is equivalent to A if one can get from B to A by a finite number of row operations |
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elementary matrix |
E is an elementary matrix if it is one row operation away from the identity matrix and if it is a square matrix |
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equivalent statements |
1. A is nonsingular 2. Ax = 0 has only the trivial solution 3. A is row equivalent to the identity matrix 4. rref(A) = the identity matrix 5. Ax = b has a unique solution 6. A is a product of elementary matrices |
