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22 Cards in this Set
- Front
- Back
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What three conditions must be met for a set of vectors V to be a subspace? |
1.) V must be closed under addition 2.) V must be closed under multiplication 3.) V must contain the zero Vector. |
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Define the null space of an m × n matrix A |
The set of vectors where Ax=0 |
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Define the column space of an m × n matrix A. |
Col A is the span of the linear combinations of A. |
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Define the kernel of a linear transformation T : V → W |
The kernel of T is the set of all vectors U within V such that T(u) = 0. |
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Define the range of a linear transformation T : V → W. |
the range of T is the set of vectors in W of the form T(x) for some x in V (in other words some vector in V imputed in the transformation will yield a vector in W) |
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Let V be a vector space. Define a basis B for V . |
B is a basis of V if B is a linearly independent set and V = Span B |
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define x in terms of cord matrix and cord vector |
x = P_b [x]_B. |
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define coordinate vector of some vector x in space V |
x = B * [x]_b were B is a basis for the space V |
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What is Rank A? |
number of pivots in A |
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using P(B |
[x]_B = P(B x in terms of B is equal to change cord matrix of c to b times c x in terms of c |
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what does the inverse of the change cord matrix from C to B equal? |
the change cord matrix of B to C |
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A scalar λ is called an eigenvalue of an n × n matrix A if . . . |
Ax = λx has a non trivial solution. |
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An eigenvector of an n × n matrix A is a nonzero vector ~xsuch that . . . |
Ax = λx for some scalar λ |
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T/F: If (A − λI)x = 0 has a nontrivial solution, then λ is an eigenvalue of A. |
True, if this equals 0 then (A − λI)x = 0 => Ax - λx=0 => Ax = λx |
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What is the Eigenspace of a vector space A? |
The eigenspace of A is equal to the set of solutions that satisfy (A-λI)x = 0, therefore it is the Nullspace of (A-λI) |
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equation for the characteristic equation (for eigen stuff) |
det(A-λI) |
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Two square matrices A and B are called similar if there exists some invertible matrixP such that |
A = PBP^-1 |
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A square matrix A is said to be diagonalizable if A is similar to a diagonal matrix such that |
A = PDP^-1 if the eigenvectors of A are linearly independent |
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Compute A^k where k is an arbitrary positive integer. |
A^k = (PDP^-1)^k => PD^kP^-1 |
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what property of a matrix A corresponds to the dimension of the eigenspace of A? (assuming A has an eigenspace) |
Dimension of the eigenspace of A is equal to the dimension of the nullspace of A, that is, the amount of free variables in A after row reduction. |
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how do you diagonalize a matrix A? |
Find the Eigen values of A, then find a basis for A that corresponds to each eigenvalue. check if the sets of bases you just found are linearly independent. If they are then P = [(set for eigen 1) (set for eigen 2) .... (set for eigen n)] and the main diagonal of D is equal to (eigen 1, eigen 2, ....., eigen n) |
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Compute [T(x)]_B. by converting the T(x) you just found in the standard basis, tothe basis B. i.e. use P^−1_B = P^−1 |
[T(x)]_B = P^−1 T(x) |
