Linear Algebra Question

Question

. Let 𝑊 be the set of all matrices of the following form with real numbers 𝑎, 𝑏, 𝑐, 𝑑.

[ 𝑎 𝑏 0

𝐴 = | 𝑏 𝑐 𝑑

0 𝑑 𝑎 ]

a) Determine whether the set 𝑊 is a subspace of 𝑀₃₃ (vector space of all 3 × 3 matrices) with the standard operations of addition and scalar multiplication.

b) Find a basis for the set 𝑊 and determine its dimension. (Give reason for your answer).

Answer 1

(a) A set W is a subspace of M₃₃ if

i) the zero matrix (of size 3 x 3) falls in W.

ii) if A and B are two matrices in W then A+B is also in W (closed under addition)

iii) if t is a real number and A is a matrix in W then aA is also in W (closed under scalar multiplication).

Let us see if i,ii,iii are true.

We start with i: Setting a=b=c=d=0, we get the zero matrix which falls in W. So i is true.

We proceed with ii: If A and B are two matrices in W, then

[ 𝑎₁ 𝑏₁ 0

𝐴 = | 𝑏 𝑐₁ 𝑑₁

0 𝑑₁ 𝑎₁ ]

and

[ 𝑎₂ 𝑏₂ 0

B = | 𝑏 𝑐₂ 𝑑₂

0 𝑑₂ 𝑎₂ ]

where a1,b1,c1,d1,a2,b2,c2,d2 are all real numbers. Adding A with B we get

[ 𝑎 𝑏 0

𝐴+B = | 𝑏 𝑐 𝑑

0 𝑑 𝑎 ]

where a = a1+a2, b=b1+b2, c=c1+c2 and d=d1+d2

So A+B is in W since it has the form of matrices in W.

Now we proceed to iii, which tells us that if we have a scalar t and matrix A then we get

[ t𝑎 t𝑏 0

tA = | t𝑏 t𝑐 t𝑑

0 t𝑑 t𝑎 ]

which is also in W.

Since we have 4 free parameters in the matrix, then the dimension of W would be 4.