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| Roots of Characteristic Equation |
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| fundamental Theorem of Algebra tells us |
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| that a polynomial of degree 'n' has at most n distinct roots |
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| homogeneous system is when |
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| 3 vectors, v1, v2, v3 are linearly independent provided that |
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| the only solution to x1v2 + x2v2+ x3v3 = 0 is the trivial solution i.e. x1=x2=x3=0 |
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| the collection of all vectors that can be written as a linear combination of v1, v2, v3. |
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| Let V be a vector space. B = {b1, b2,...bn} is a basis for V if |
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| b1, b2, b3 are linearly independent and B spans V |
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| Subspaces V has three properties: |
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| 0 vector is in V; if a and b are in V so is a + b; if a is in V then so is ca where c is any scalar |
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NulA - all vectors x such that Ax = 0;
ColA - all vectors that can be written as a linear combination of the columns of A |
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| Matrix multiplication is _____ commutative |
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| Linear transformation L has 3 properties: |
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| L(0vector) = 0 vector; L(v1 + v2) = L(v1) + L(v2); L(cv1) = cL(v1) |
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| dimensions of a subspace equal |
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| number of vectors in a basis |
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| fundamental theorem of linear algebra is: |
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| the roots to det(A - λI) = 0 |
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| Eigenvectors are from (come from) |
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| the nullspace of (A - λI)x = 0 |
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| in equation A = PDPT columns of P are: |
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| A property of a diagonalizable matrix |
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| Raising A to a power is simplified - AK = PDKP-1 |
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| A matrix U has orthonormal columns if |
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| if and only if U^t x U = I |
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| If U has orthonormal columns then some properties of U are: |
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U preserves distances and dot products:
aka || Ux|| = ||x||; Ux • Uy = x • y Ux • Uy = U0vector if and only if x "dot" y = 0 |
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| Gram - Schmidt Algorithm is |
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v1=x1 v2 = x2 - ((x2 • v1)/(v1 • v1)) x v1
v3 =x3-((x3 • v1)/(v1 • v1)) x v1 - (x3 • v2)/(v2 • v2) x v2 |
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y = ((y • U1)/(U1 • U1))x U1 +......
((y • Up)/Up • Up) x Up |
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| Invertible Matrix Theorem has following properties: (either all true or all false) |
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1) A is invertible
2) A is row equivalent to the nxn identity matrix 3)A has n pivot positions
4)Ax=0vector has only the trivial solutions 5)columns of A are linearly independent
6)The linear transformation x to Ax is 1-to-1
7)Ax = b is always consistent
8)Columns of A span R^n
9)the linear transformation x to Ax maps R^n onto 1
10) There exists Cnxn such that CA = I
11) there exists Dnxn such that AD = I
12) AT is invertible matrix
13) det(A)/=0 |
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