1. If 12 n is a basis for n and is an invertible matrix, show that 12n is also a basis for n. 2. If 123 are linearly independent, for what values of c are the vectors 2132 and 13 linearly...

1. If

<sub>1

2


n</sub>
is a basis for

ⁿ
and

is an invertible

matrix, show that

<sub>1

2

n</sub>
is also a basis for

ⁿ.

2. If

<sub>1

2

3</sub>
are linearly independent, for what values of c are the vectors

<sub>2

1

3

2</sub>
and

<sub>1

3</sub>
linearly independent

3. Prove that if

is an

matrix, rank

rank

^T

4. Given

matrices

and

prove that rank

min
rank

rank

Does


give you a portion of the required result
The result of Problem 3.14 says that rank


rank


^T