Practice Quiz

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1. Prove the statement

where { : such that = 2}.

Prove that every spanning set of a finite-dimensional vector space contains a basis.

Give an example of a non-diagonalizable linear operator on a finite-dimensional vector space.

Show that every positive definite matrix has a square root. Give an example of a symmetric matrix with no square root.

Prove that every matrix "$ has a singular value decomposition.

Show that a metric space is compact (in the topological/open cover definition) iff it is sequentially compact.

Rigorously define the Riemann integral of real-valued function on a compact interval.

Show that the function

1() =1, 0,

is not Riemann integrable.