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## Non-degenerate form induces adjoint linear operators

Solution to Linear Algebra Hoffman & Kunze Chapter 9.2 Exercise 9.2.9 Solution: We can always define an inner product on a finite-dimensional vector space $V$. Hence we shall assume that…

## Relation between non-degenerate forms and linear functionals

Solution to Linear Algebra Hoffman & Kunze Chapter 9.2 Exercise 9.2.8 Solution: We can always define an inner product on a finite-dimensional vector space $V$. Hence we shall assume that…

## Form is left non-degenerate if and only if it is right non-degenerate

Solution to Linear Algebra Hoffman & Kunze Chapter 9.2 Exercise 9.2.7 Solution: Call the form $f$ right non-degenerate if $0$ is the only vector $\alpha$ such that $f(\beta,\alpha)=0$ for all…

## Form is non-degenerate if and only if the associated linear operator is non-singular

Solution to Linear Algebra Hoffman & Kunze Chapter 9.2 Exercise 9.2.6 Solution: If $f$ is non-degenerate, we show that $T_f$ is non-singular. Suppose $T_f\alpha=0$ for some $\alpha\in V$, then we…