30 The set of all n x n diagonal matrices 32 The set of all

30. The set of all n x n diagonal matrices 32. The set of all n x n diagonal matrices A that commute with a given matrix B ; that is, AB = BA

Solution

22) The set of all negative functions f(x)<0 is not a subspace

      the set of all functions whose output is always is negative or non-positive is not a subspace.

because it is not closed under scalar multiplication

for example f(x)=-x²

if we multiply this function by -1 it becomes x²

which is not in the set

so the set of all negative functions f(x)<0 is not a subspace.

27)

The set of all functions such that f(0)=0 is a subspace.

suppose that f(0)=0 and g(0)=0

so that both f and g are in the set

then (f+g)(0)=f(0)+g(0)

                  =0+0

                  =0

so f+g is in the set

the set is closed under addition.

Let k be any scalar

then k*f(0)=k*(0) [f(0)=0]

              =0

so k*(f) is also in the set

so the set is closed under scalar multiplication.

so it is closed under both operations.

so it is a subspace.

30) the set of all n*n diagonal matrices is a subspace.

since M₀ is a diagonal matrix.

diagonal matrix means

A square matrix is a diagonal if all its off diagonal entries are zero and diagonal entries might be zero or not.

32)The set of all n*n matrices A that commute with a given matrix B that is BA=AB is a subspace

let A be the fixed matrix M_nxn and S be the set all matrices that commute with A i.e{B/AB=BA

let I be the n*n identity matrix then AI=IA=A

so I belongs to S and S is non empty.

let B₁,B₂ belongs to S

then (B₁+B₂)A=B₁A+B₂A=AB₁+AB₂=A(B₁+B₂)

so (B₁+B₂) S and S is closed under addition.

let B S and M

then (B)A=(I)BA=(I)(AB)=A(I)B=A(B)

so (BS) and S is closed under scalar multiplication.

So S is a subspace on M_n*n

So finally The set of all n*n matrices A that commute with a given matrix B that is BA=AB is a subspace

33)The set of all n*n singular matrices is not a subspace

because it does not satifies the property closed under addition.

for example i am taking two matrices [1 0

                                                      0 0]

and [0 0

       0 1] are singular because their sum is I which is non singular

there fore the set of all n*n singular matrices is not a subspace.

34) the set of all n*n invertiable matrices is not a subspace

since the set is not closed under vector addition and scalar multiplication.

let I is in the set i.e identity matrix

but 0*I =0 zero matrix is clearly not in the set (not being invertiable)

so this set is not closed under scalar multiplication.

similarly by adding any invertiable matrix to its opposite(also invertiable)

so this set is not closed under addition and as well as scalar multiplication.

35)Yes set of all n*n matrices whose entries add up to zero is a subspace because it is easy

to chech that diagonal matrices are closed under addition and scalar multiplication.