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Part I: True/False For each of the following mark as (T) true or (F) false. If you marked as false, provide a counterexample. If

`A`

and

`B`

are

`n\times n`

lower triangular matrices, then

`AB`

is also lower triangular. If

`A`

and

`B`

are

`n\times n`

symmetric matrices, then

`AB`

is symmetric. If

`A`

is a

`4\times 2`

matrix, then

`Ax=0`

has at least two free variables.

`tr(ABC)=tr(BAC)`

for an

`n\times n`

matrices

`A,B`

, and

`C`

. If

`W`

is a subspace of a vector space

`V`

and

`B`

is a basis for

`W`

, then

`B`

can be extended to a basis for

`V`

.

`tr(AB)=tr(A)tr(B)`

for an

`n\times n`

matrices

`A,B`

, and

`C`

. If

`W`

is a subspace of a vector space

`V`

and

`B`

is a basis for

`V`

, then

`B`

can be restricted to a basis for

`W`

. For any

`m\times n`

matrices

`A`

and

`B`

,

`B=EA for some invertible ELongleftrightarrowNS(A)=NS(B). `