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31 Cards in this Set
- Front
- Back
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State: The complex spectral theorem.
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What is the Pythagorean theorem?
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State: The complex spectral theorem.
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State the Cauchy Schwarz inequality.
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State the triangle inequality (using norms).
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State the parallelogram equality.
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State: The generalized Pythagorean identity.
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Let (v1,...vn) be an orthogonal basis of V and let v=a1v1+...+anvn. How do you compute ai?
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How do you calculate u3 in the Gram Schmidt process?
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Define: Unitary matrix.
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How do you calculate the projection of v2 on v1?
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Let (V, <,>) be an inner product space with orthonormal basis B=(e1, ...., en).Let v be an element of V.
1. What are the Fourier coefficients? 2. What is the Fourier expansion? |
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What is Parseval’s identity?
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State: Bessel’s inequality.
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Give the sum formula for the projection of v on u.
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State: The Riesz representation theorem.
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Define: The adjoint of T.
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Define: The dual space of V.
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Every operator, T, on a finite-dimesnsional, nonzero, complex vector space V has an _____.
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eigenvalue
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Suppose TL(v) has an upper triangular matrix wrt some basis of V. Then what conditions do the diagonal entries have to fulfill in order for the matrix to be invertible?
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The diagonal entries must be nonzero.
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Suppose V is a complex vector space and T is an element of L(v). Then there exists a basis B of V such that m_B(T)...
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is upper triangular.
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Let TeL(v) and let U be a subspace of V. What does it mean for U to invariant under T (or T-invariant)?
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T(u) is an element of U.
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Suppose TeL(v)has an upper triangular matrix wrt some basis V. What are the eigenvalues of T?
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The diagonal entries of the matrix.
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Define: Self-adjoint.
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T=T*
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Every eigenvalue of a self adjoint operator is...
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real.
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Let V be a complex IPS, and let TL(v). Then T is self adjoint iff
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<T(v),v>eR
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If T:VV, and T is self adjoint (ie. T=T*), then there is an orthonormal basis of eigenvectors and all eigenvalues are ...
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real
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When is TeL(v) normal?
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When TT*=T*T.
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Supposse TeL(v) is normal. If veV is an eigenvector of T with eigenvalue a, then v is also an eigenvalue of T* with eigenvalue ___
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conjugate of a
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If TL(v) is normal, then eigenvectors of T corresponding to distinct eigenvalues are
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orthogonal
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TL(v) has a diagonal matrix representation (wrt some basis V) iff ...
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V has a basis consisting of eigenvectors of T.
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How can you rewrite x^2+ax+b?
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