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18 Cards in this Set
- Front
- Back
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What is the equation must an eigenvalue satisfy?
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(A-lambda*In)x=0
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How do you find an eigenvalues?
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1) Plug lambda into the diagonal of the matrix
2) Find the determinant 3) Set it to zero (characteristic equations) 4) Solutions to the equation are the eigenvalues |
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What is a nullspace of a matrix?
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The solution space of equations Ax=0
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What is a kernel of a linear transformation?
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For linear transformation using matrix, A, the kernel is the solution space of equations Ax=0
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What is the dimension of a matrix?
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The number of vectors in the basis (and a basis consists only of linerally independent vectors)
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How can you determine vectors are linearlly independent?
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The only solution to Ax=0 is the zero vector.
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How can you test for matrix invertibility?
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1) Rank of the matrix must equal the number of rows, columns (no zero rowspaces, columnspaces)
2) Columns must be linerally independent 3) Determinant must not equal 0 |
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What is algebraic multiplicity?
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The algebraic multiplicity of an eigenvalue is the number of times an eigenvalue appears in the list of total eigenvalues. Put another way, it is the highest value of 'k' if (t-c)^k makes up the characteristic equation.
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What is the geomtric multiplicity?
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The geometric multiplicity of an eigenvalue is the dimension of the eigenspace corresponding to the eigenvalue.
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What is greater - the algebraic or geomteric multiplicity of an eigenvalue?
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The algebraic multiplicity is always greater than or equal to the geometric multiplicity.
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How can you determine if a matrix is diagonalizable?
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If and only if, for every eigenvalue, the geometri multiplicity is equal to the algebraic multiplicity.
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What is the range of a linear transformation?
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If Ax: Rn -> Rm then the range is the column space of A
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What are the conditions for a vector subspace? (Subset W of Rn)
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1. The zero vector is in W (0*W=0)
2) If vector u and v are in W, then u+v must be in W 3) If k is any scalar and vector u is in W then k*u must be in W. |
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How can you determine if a set of vectors is the basis?
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- Transform the column vectors into row vectors
- Put them in a matrix A - If the zero vector is the only solution to Ax=0 then the vectors form the basis |
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How can you tell if a vector '' is in a subspace spanned by a set of vectors v?
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- Combine the vectors in S into a matrix
- Augment the matrix with v - If there is a solution, then v is in S Meaning, there must be a solution to to Sx=b |
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What is the null space of a matrix?
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The nullspace of matrix A is the solution set of Ax=0
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What is the inner product of two vectors 'u' and 'v'?
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The inner product of 'u' and 'v' (or u dot v) is u12 + u2v2+ ... unvn.
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What is the length of vector x or ||x||?
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The length of x is the square root of the sum of squares of the vector values.
||x||=sqrt(x1^2 + x2^2 +... xn^2) |
