Podcast
Questions and Answers
Is $E_
ho$ a subspace of U for a linear operator $L: U o U$ and scalar $
ho$ in K?
Is $E_ ho$ a subspace of U for a linear operator $L: U o U$ and scalar $ ho$ in K?
- Yes (correct)
- No
The matrix $M -
ho I$ is invertible if and only if $
ho$ is not an eigenvalue of $M$.
The matrix $M - ho I$ is invertible if and only if $ ho$ is not an eigenvalue of $M$.
True (A)
If $B$ or $C$ are zero in matrix $M := egin{pmatrix} A & B \ C & D \ \ \ \ \end{pmatrix}$, what is true about the characteristic polynomial?
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?
What is the algebraic multiplicity of an eigenvalue?
What is the minimal polynomial of the diagonal matrix $diag(a_1,..., a_n)$?
What is the minimal polynomial of the diagonal matrix $diag(a_1,..., a_n)$?
If $
ho$ is an eigenvalue of $L$, then $
ho^{-1}$ is also an eigenvalue of $L^{-1}$.
If $ ho$ is an eigenvalue of $L$, then $ ho^{-1}$ is also an eigenvalue of $L^{-1}$.
What happens to the characteristic polynomial $F(t)$ of matrix $M$ when $det(-M) = 0$?
What happens to the characteristic polynomial $F(t)$ of matrix $M$ when $det(-M) = 0$?
What does the Fundamental Theorem of Algebra state regarding a degree $n$ polynomial?
What does the Fundamental Theorem of Algebra state regarding a degree $n$ polynomial?
Prove that $
ho^n$ is an eigenvalue of $L^n$ if $
ho$ is an eigenvalue of $L$. What is the expression?
Prove that $ ho^n$ is an eigenvalue of $L^n$ if $ ho$ is an eigenvalue of $L$. What is the expression?
The eigenvectors corresponding to distinct eigenvalues are linearly independent.
The eigenvectors corresponding to distinct eigenvalues are linearly independent.
For distinct eigenvalues $
ho_1$ and $
ho_2$, the intersection of their eigenspaces contains only the zero vector.
For distinct eigenvalues $ ho_1$ and $ ho_2$, the intersection of their eigenspaces contains only the zero vector.
An orthogonal set of non-zero vectors in an inner product space is always linearly independent.
An orthogonal set of non-zero vectors in an inner product space is always linearly independent.
For a subset $S$ of an inner product space, prove that $S
eq (S^ot)^ot$. What does this imply?
For a subset $S$ of an inner product space, prove that $S eq (S^ot)^ot$. What does this imply?
If $S_1 ackslash S_2$, what is true about their orthogonal complements?
If $S_1 ackslash S_2$, what is true about their orthogonal complements?
The angle between any two non-zero elements $u$ and $v$ of an inner product space is zero if they are linearly independent.
The angle between any two non-zero elements $u$ and $v$ of an inner product space is zero if they are linearly independent.
What can be said about the existence of an orthonormal basis for a subspace $W$ within an inner product space $V$?
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
-
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.
-
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.
-
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 ).
-
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.
-
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 ).
-
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.
-
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.
-
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 ).
-
Two nonzero elements ( u ) and ( v ) of an inner product space are linearly independent if the angle between them is non-zero.
-
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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