HW 2.7

A={{0,0,2,4},{2,-4,1,0},{-3,6,2,7}};
u={0,0,0,0};
v={-2,0,4,-2};

A.u

A.v

NullSpace[A]

HW 2.6

A={{2,2,4},{-4,-4,-8},{0,-3,-3}};
b={8,-16,12};

NullSpace[A]

p=LinearSolve[A,b]

A.p

RowReduce[A]

Inverse[A]

HW 4.2

RowReduce[{{1,-1,2},{-2,0,1},{3,-1,1}}]

not injective since column 3 has no pivot, not surjective since there is a zero row

HW 4.6

A={{2,1,-4},{3,-1,1}}; B = {{1,1},{-3,4},{2,0}};

A.B

B.A

HW 5.1

b=. 
A={{a,b},{c,d}}

Inverse[A]

HW 3.1

v={{0,-1,4},{2,1,3},{1,0,-2}};
MatrixForm[Transpose[v]]

NullSpace[Transpose[v]]

Null space is trivial, hence the vectors are linearly independent.