How can I show that all symmetric 2x2 matrices form a subspace of the vector space M2 of all square 2x2..?

Question

matrices?

I also need to figure out the basis for this subspace. Also what does it mean when I need to solve for the same concept but in skew-symmetric matrix?

Answer 1

To show the property of being a subspace you need only show that it's closed under scalar multiplication and addition of vectors.



Suppose a symmetric 2x2 matrix M.



[a b]

[b c]



Suppose we take a scalar from the field the vector space is over called s. Then s*M is equivalent to:



[s*a s*b]

[s*b s*c]



Clearly this matrix is also symmetric, so our first condition is satisfied.



Suppose another matrix M' defined as follows:



[d e]

[e f]



Then M+M' is in the space if the sum is symmetric.



[a+d b+e]

[b+e c+f]



which is clearly a diagonal matrix.



This demonstrates that the set of symmetric 2x2 matrices is a subspace of the set of all 2x2 matrices. As for your question about skew symmetric, apply the same methodology to skew symmetric matrices and see if the result is true. A skew symmetric matrix is one in which the transpose of the matrix is the same as its negative. In other words, it's a symmetric matrix in which the signs of the elements flipped across the diagonal are also flipped. For instance:



[2 3]

[-3 4]



is skew symmetric.

Answer 2

The set of S, 2x2 symmetric matrices must satisfy the following conditions in order to be a subspace the vector space M of all 2x2 matrices

A) S ⊂ M

B) S is a vector space



In practical terms, this can be shown by demonstrating that S satisfies the following four conditions. Keep in mind that S is the set of matrices of the form

⌈a...b⌉

⌊b...c⌋



1) S ⊂ M : Trivially true

2) 0 ∈ S

⌈0...0⌉

⌊0...0⌋ is symmetric, so 2 holds

3) If u ∈ S and v ∈ S, then u + v ∈ S

Let u =

⌈a...b⌉

⌊b...c⌋

and v =

⌈d...e⌉

⌊e...f⌋

u + v =

⌈a+d...b+e⌉

⌊b+e...c+f⌋ is symmetric, so 3 holds

4) If u ∈ S and g is a scalar, then gu ∈ S

gu=

⌈ga...gb⌉

⌊gb...gc⌋ which is symmetric, so 4 holds

Therefore S is a subspace of M



u =

a⌈1...0⌉ +

..⌊0...0⌋

b⌈0...1⌉+

..⌊1...0⌋

c⌈0...0⌉

..⌊0...1⌋.

So these three matrices form a basis for S



Similar reasoning holds for the skew symmetric case

Answer 3

Consult Wikipedia for the definition of vector space. A VS requires 8 axioms which are comparatively easy to verify in your example.