Determine whether the subset of ${M}_{n,n}$ is a subspace of ${M}_{n,n}$ with the standard operations. Justify your answer.

The set of all$n\times n$ matrices whose entries sum to zero

The set of all

CoormaBak9
2020-11-10
Answered

Determine whether the subset of ${M}_{n,n}$ is a subspace of ${M}_{n,n}$ with the standard operations. Justify your answer.

The set of all$n\times n$ matrices whose entries sum to zero

The set of all

You can still ask an expert for help

Isma Jimenez

Answered 2020-11-11
Author has **84** answers

Step 1

Given that,

The set of all$n\times n$ matrices whose entries sum to zero is a subset of ${M}_{n,n}$

A nonempty subset W of vector space V is a subspace if it is closed under addition and scalar multiplication.

That is, if u, v in W then u +v lies in W.

If a is any scalar then au also in W.

Let W is the set of all$n\times n$ matrices whose entries sum to zero.

As n by n zero matrix whose entries sum to zero.

Thus W is non-empty.

Let A and B are two n by n matrix such that all entire sum add up to zero.

${a}_{ij},\dots ,i,j=1,2,3,\dots n$ denotes the entries in the matrix A

${b}_{ij},\dots ,i,j=1,2,3,\dots n$ denotes the entries in the matrix B

Thus,$\sum _{i,j=1}^{n}{a}_{ij}=0$

$\sum _{i,j=1}^{n}{b}_{ij}=0$

Step 2

Consider the sum of entries in A+B

$\sum _{i,j=1}^{n}{a}_{ij}+{b}_{ij}$

By using summation property,

$\sum _{i,j=1}^{n}{a}_{ij}+\sum _{i,j=1}^{n}{b}_{ij}$

It gives,

0 +0 =0

Thus,$\sum _{i,j=1}^{n}{a}_{ij}+{b}_{ij}=0$

Therefore,

All entries in A+B has sum zero.

A+B lies in W.

Step 3

Now take any scalar u in real number.

Let A is in W.

$\sum _{i,j=1}^{n}{a}_{ij}=0$

We have to show that uA is in W.

Take the sum of all entries in uA

$\sum _{i,j=1}^{n}u{a}_{ij}$

$u\sum _{i,j=1}^{n}{a}_{ij}$

$u(0)=0$

Thus, all entries in uA have sum zero.

Thus, uA lies in W.

Thus W is a subspace of${M}_{n,n}$

Therefore, the set of all$n\times n$ matrices whose entries sum to zero is a subspace of ${M}_{n,n}$ .

Given that,

The set of all

A nonempty subset W of vector space V is a subspace if it is closed under addition and scalar multiplication.

That is, if u, v in W then u +v lies in W.

If a is any scalar then au also in W.

Let W is the set of all

As n by n zero matrix whose entries sum to zero.

Thus W is non-empty.

Let A and B are two n by n matrix such that all entire sum add up to zero.

Thus,

Step 2

Consider the sum of entries in A+B

By using summation property,

It gives,

0 +0 =0

Thus,

Therefore,

All entries in A+B has sum zero.

A+B lies in W.

Step 3

Now take any scalar u in real number.

Let A is in W.

We have to show that uA is in W.

Take the sum of all entries in uA

Thus, all entries in uA have sum zero.

Thus, uA lies in W.

Thus W is a subspace of

Therefore, the set of all

Jeffrey Jordon

Answered 2022-01-30
Author has **2262** answers

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