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problems and solutions here.…

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solutions to 2.14

Exercise:
Let \(V\) be the space of real polynomial functions of degree at most \(3\). Consider the quadratic form \(q_1:V \to \mathbb{R}\) given by $$ q_1(f):=|f(-1)|^2+|f(0)|^2+|f(1)|^2. $$ Is this form positive definite?
Consider another form \(q:V\to \mathbb{R}\) given by $$ …

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2. 14 quadratic forms

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}\def\Field{\mathbb{F}}\def\Fun{\mathbf{Fun}}\def\e{\mathbf{e}}
\def\f{\mathbf{f}}\def\bv{\mathbf{v}}\def\i{\mathbf{i}}
\def\eye{\left(\begin{array}{cc}1&0\\0&1\end{array}\right)}
\def\bra#1{\langle #1|}\def\ket#1{|#1\rangle}\def\j{\mathbf{j}}\def\dim{\mathrm{dim}}
\def\ker{\mathbf{ker}}\def\im{\mathbf{im}}
\def\tr{\mathrm{tr\,}}
\def\braket#1#2{\langle #1|#2\rangle}
\)

1. Bilinear forms are functions \(Q:U\times U\to k \) that depend on each of the arguments linearly.

Alternatively, one can think of them as the linear operators
\[
A:U\to U^*, …

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solutions to 2.9

Exercise: consider the mapping that takes a quadratic polynomial \(q\) to its values at \(0,1,2\) and \(3\). Find the normal form of this operator.
Solution: let \(V\) denote the space of quadratic polynomials with the standard basis \(\{1,x,x^2\}\), …

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2.9 normal forms

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}\def\Field{\mathbb{F}}\def\Fun{\mathbf{Fun}}\def\e{\mathbf{e}}
\def\f{\mathbf{f}}\def\bv{\mathbf{v}}\def\i{\mathbf{i}}
\def\eye{\left(\begin{array}{cc}1&0\\0&1\end{array}\right)}
\def\bra#1{\langle #1|}\def\ket#1{|#1\rangle}\def\j{\mathbf{j}}\def\dim{\mathrm{dim}}
\def\ker{\mathbf{ker}}\def\im{\mathbf{im}}
\def\tr{\mathrm{tr\,}}
\def\braket#1#2{\langle #1|#2\rangle}
\)
1. We know that an operator can be represented as a matrix if you fix the basis. Changing the basis changes the matrix. One can try to make the matrix …

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solutions to 2.7

Exercise: find the product of rotations by \(\frac{\pi}{2}\) around \(x, y\) and then \(z\) axes.
Solution: \( \newcommand{Rot}[1]{\overset{R_#1}{\longrightarrow}} \) Let \(R_x,R_y\) and \(R_z\) represent the rotation by \(\frac{\pi}{2}\) matrices about the \(x,y\) and \(z\) axes respectively, as defined …

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2.7 functions of operators

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}\def\Field{\mathbb{F}}\def\Fun{\mathbf{Fun}}\def\e{\mathbf{e}}\def\f{\mathbf{f}}\def\bv{\mathbf{v}}\def\i{\mathbf{i}}
\def\eye{\left(\begin{array}{cc}1&0\\0&1\end{array}\right)}
\def\bra#1{\langle #1|}\def\ket#1{|#1\rangle}\def\j{\mathbf{j}}\def\dim{\mathrm{dim}}
\def\braket#1#2{\langle #1|#2\rangle}
\def\ker{\mathbf{ker}}\def\im{\mathbf{im}}
\def\e{\mathcal{E}}
\def\tr{\mathrm{tr}}
\)

1.
A linear operator \(A:U\to U\) that maps a space into itself is called endomorphism. Such operators can be composed with impunity. In elevated language, they form an algebra. (It …

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solutions to exercises 1.26

Exercise: Find the determinants of Jacobi matrices with \(a=1,bc=-1\) and \(a=-1,bc=-1\).
Solution: First assume \(a=1,bc=-1\), and use the regression obtained in the lecture notes, i.e $$j_{k+1}=aj_k-bcj_{k-1} = j_k+j_{k-1}$$

Those are Fibonacci numbers with initial conditions obtained by the first two …

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1.26 determinants

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}\def\Field{\mathbb{F}}\def\Fun{\mathbf{Fun}}\def\e{\mathbf{e}}
\def\f{\mathbf{f}}\def\bv{\mathbf{v}}\def\i{\mathbf{i}}
\def\eye{\left(\begin{array}{cc}1&0\\0&1\end{array}\right)}
\def\bra#1{\langle #1|}\def\ket#1{|#1\rangle}\def\j{\mathbf{j}}\def\dim{\mathrm{dim}}
\def\ker{\mathbf{ker}}\def\im{\mathbf{im}}
\)

1. Determinant is a function of square matrices (can be defined for any operator \(A:U\to U\), but we’ll avoid this abstract detour). It can be defined as a function that has the following …

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1.24 linear operators

\(\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}\def\Field{\mathbb{F}}\def\Fun{\mathbf{Fun}}\def\e{\mathbf{e}}
\def\f{\mathbf{f}}\def\bv{\mathbf{v}}\def\i{\mathbf{i}}
\def\eye{\left(\begin{array}{cc}1&0\\0&1\end{array}\right)}
\def\bra#1{\langle #1|}\def\ket#1{|#1\rangle}\def\j{\mathbf{j}}\def\dim{\mathrm{dim}}
\def\ker{\mathbf{ker}}\def\im{\mathbf{im}}
\)

1. Linear mappings, or linear operators are just linear functions taking values in a linear space:
\[
A:U\to V,\ \mathrm{where\ both\ } U\ \mathrm{and\ } V\ \mathrm{ are\ linear\ spaces.}
\]
The same …

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