Questions and Answers
Is $E_ ho$ a subspace of U for a linear operator $L: U o U$ and scalar $ ho$ in K?
The matrix $M - ho I$ is invertible if and only if $ ho$ is not an eigenvalue of $M$.
True
If $B$ or $C$ are zero in matrix $M := egin{pmatrix} A & B \ C & D \ \ \ \ \end{pmatrix}$, what is true about the characteristic polynomial?
It is the product of the characteristic polynomials of $A$ and $D$.
What is the algebraic multiplicity of an eigenvalue?
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What is the minimal polynomial of the diagonal matrix $diag(a_1,..., a_n)$?
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If $ ho$ is an eigenvalue of $L$, then $ ho^{-1}$ is also an eigenvalue of $L^{-1}$.
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What happens to the characteristic polynomial $F(t)$ of matrix $M$ when $det(-M) = 0$?
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What does the Fundamental Theorem of Algebra state regarding a degree $n$ polynomial?
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Prove that $ ho^n$ is an eigenvalue of $L^n$ if $ ho$ is an eigenvalue of $L$. What is the expression?
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The eigenvectors corresponding to distinct eigenvalues are linearly independent.
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For distinct eigenvalues $ ho_1$ and $ ho_2$, the intersection of their eigenspaces contains only the zero vector.
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An orthogonal set of non-zero vectors in an inner product space is always linearly independent.
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For a subset $S$ of an inner product space, prove that $S eq (S^ot)^ot$. What does this imply?
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If $S_1 ackslash S_2$, what is true about their orthogonal complements?
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The angle between any two non-zero elements $u$ and $v$ of an inner product space is zero if they are linearly independent.
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What can be said about the existence of an orthonormal basis for a subspace $W$ within an inner product space $V$?
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Study Notes
Linear Algebra Proofs Study Notes
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A linear operator ( L: U \to U ) has a subspace ( E_\lambda ) corresponding to eigenvalue ( \lambda ) if for all ( u_n \in E_\lambda ), the equation ( L(au_1 + bu_2) = \lambda(au_1 + bu_2) ) holds.
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The matrix ( M - \lambda I ) is invertible if and only if ( \lambda ) is not an eigenvalue of ( M ). If ( \lambda ) is an eigenvalue, then ( \text{det}(M - \lambda I) = 0 ), which makes ( M ) non-invertible.
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For a block matrix ( M = \begin{pmatrix} A & B \ C & D \end{pmatrix} ), if either ( B ) or ( C ) is zero, the characteristic polynomial of ( M ) is the product of the characteristic polynomials of ( A ) and ( D ).
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The algebraic multiplicity of an eigenvalue is at least 1 as indicated by the characteristic polynomial ( p(t) = (t_1 - a_1)^{m_1}...(t_n - a_n)^{m_n} ). If ( m_1 < 0 ), ( a_1 ) is not an eigenvalue, and if ( m_1 = 0 ), it reduces to 1.
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The minimal polynomial of a diagonal matrix ( \text{diag}(a_1,..., a_n) ) is given by ( (t - a_1)...(t - a_n) ). The irreducibility of the minimal polynomial entails that if an irreducible polynomial divides the minimal polynomial, it must also divide the characteristic polynomial.
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In a vector space ( V ) with dimension ( n ), if ( L: V \to V ) has an eigenvalue ( \lambda ) with multiplicity ( k ), the geometric multiplicity of any other eigenvalue of ( L ) is limited to ( n - k ).
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The sum of all algebraic multiplicities of eigenvalues for a matrix or transformation gives the dimension of the space. The inverse of ( \lambda ) will also be an eigenvalue if ( \lambda ) is an eigenvalue of ( L ).
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For an ( n \times n ) matrix ( M ), the characteristic polynomial ( F(t) ) is defined as ( f(t) = \text{det}(tI - M) ). If ( F(0) = \text{det}(-M) = 0 ), then ( M ) is singular.
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The fundamental theorem of algebra asserts that any degree ( n ) polynomial has ( n ) complex roots, thereby confirming that the sum of the algebraic multiplicities of eigenvalues equals ( n ).
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If ( L: V \to V ) and ( \lambda ) is an eigenvalue of ( L ), it can be shown that ( \lambda^n ) is also an eigenvalue of ( L^n ).
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A set of nonzero eigenvectors ( {v_1, ..., v_k} ) corresponding to distinct eigenvalues is linearly independent.
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For distinct eigenvalues ( \lambda_1 ) and ( \lambda_2 ) of ( L ), the intersection ( E_{\lambda_1} \cap E_{\lambda_2} = {0} ) holds true, meaning they share no common nonzero vectors.
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In an inner product space, an orthogonal set of non-zero vectors is guaranteed to be linearly independent.
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For any subset ( S ) of an inner product space, ( S \subseteq (S^\perp)^\perp ) can be proven. If ( w \in S ) and is orthogonal to ( S^\perp ), then ( w ) belongs to ( (S^\perp)^\perp ).
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For subsets ( S_1 ) and ( S_2 ) of a vector space ( V ) where ( S_1 \subseteq S_2 ), it follows that ( S_2^\perp \subseteq S_1^\perp ).
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Two nonzero elements ( u ) and ( v ) of an inner product space are linearly independent if the angle between them is non-zero.
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In an inner product space ( V ) of dimension ( n ) and subspace ( W ) of dimension ( k ), it can be established that an orthonormal basis for ( V ) exists, which includes a basis for ( W ).
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Test your understanding of key linear algebra concepts with these flashcards focused on proofs. Challenge yourself on topics like eigenvalues, linear operators, and matrix properties. Perfect for students looking to deepen their knowledge in linear algebra.
