Vector Spaces: Theorems Part 4 - Example Solved Problems

Vector Spaces: Theorems Part 4

Example Problems

Example 20

Verify whether the matrix  generate M_2×2 (R) or not?

Solution:

Here a₁₁, a₁₂, a₂₁ and a₂₂ ∈ F = R.

By comparing on both sides we have

a+b+c=α₁₁    ……… (1)

a+b+d=a₁₂    ……… (2)

a+c+d=a₂₁    ……… (3)

b+c+d=a₂₂    ……… (4)

From equations (1), (2), (3) and (4), since an arbitrary matrix A in M_2×2(R) can be expressed as a linear combination of the four given matrices.

 The given matrices generate M_2×2 (R).

Example 21

Verify the vectors (1, 1, 0), (1, 0, 1) and (0, 1, 1) generate F³ or not.

Solution:

Let us consider (x, y, z) ∈ F³ and a, b, c ∈ F

Let (x, y, z) = au + bv + cw

(x, y, z) = a (1, 1, 0) + b (1, 0, 1) + c (0, 1, 1)

= (a, a, 0) + (b, 0, b) + (0, c, c)

(x, y, z) = (a + b, a+c, b+c)

a+b=x        ………… (1)

a+c=y        ………… (2)

b+c=z        ………….(3)

Equation (2) ‒ (3) gives, a−b = y−z             ……….(4)

Equation (1) + (4) gives, a+b+a−b = x+y−z

⇒ 2a=x+y¬z              ……….(5)

From (5); a = 1/2(x + y −z]

From (1) b = x−a

 b = x – ½ = [ x + y − z] = ½ [ 2x − x − y+z]

 b = 1/2 [x−y+z]           ... (6)

From equation (2); a+c=y

c=y−a

 c = y − 1/2[x+y−z]

= ½ [2y−x−y+z]

 c = ½ [y−x+2]             .. (7)

  (x, y, z) = ½ (x + y − z](1, 1, 0) + ½ (x − y + z](1, 0, 1) + ½(y−x + 2](0, 1, 1)

The given vectors generate F³.

Example 22

Let V=R³; S₁ = {(1, 0, 0) } ; S₂ = { (1, 0, 0), (2, 2, 0)} then prove that L(S₁)  L(S₂).

Solution:

L (S₁) = { α (1, 0, 0) / α ∈ F}

L (S₂) = { α (1, 0, 0) + β(2, 2, 0) / α, β ∈ F}

L(S₁)  L(S₂)

{ L(S₁) = L(S₂) if β=0 }

Note:

If one vector can be represented as the scalar multiplication of another vector (ie) v₁ = α v₂, then the linear span of V will be equal to linear span of v₂.

Example 23

Let V=R³, W₁ = {(x, x, x)/x ∈ R} and W₂ = {(0, y, z)/y, z ∈ R} are two subspaces of V, then prove that V=W₁  W₂

Solution:

 Clearly W₁ ∩ W₂ = {0}

If (x, y, z) ∈ V then we can write

 (x, y, z) = (x, x, x) + (0, y − x, z − x) ∈ W₁ + W₂.

  V=W₁ + W₂. Hence V is the direct sum of W₁ and W₂.

(ie) V = W₁  W₂

Example 24

Let V=R³, W₁ = {(x, y, 0) / x, y ∈ R} and W₂ = {(0,y,z) /y, z ∈ R } are clearly subspaces of V. But prove that V is not a direct sum of W₁ and W₂.

Solution:

Geometrically W₁ and W₂ represent set of all points in the xy plane and set of all points in yz plane respectively.

  W₁ ∩ W₂ = set of all points in y axis.

 = {(0, y, 0)} ≠ {0}.

  V is not the direct sum of W₁ and W₂.

Example 25

Show that if  then the span of {M₁, M₂, M₃} is the set of all symmetric 2×2 matrices.

Solution:

Since all the matrices are symmetric, every matrix in their span will be symmetric, hence we have to show that every symmetric matrix is in their span. Then every 2×2 symmetric matrix has the form of M =  and since we can write these matrices as M=aM₁+cM₂+bM₃ and in the span of { M₁, M₂, M₃ }.

Example 26

Show that the matrices  generate M_2×2(R).

Solution:

By comparing on both sides a = a₁₁, b=a₁₂, c=a₂₁ and d=a₂₂.

  The given matrices generate M_2×2(R).

Example 27

Show that P_n(F) is generated by {1,x,x², x³,...xⁿ}

Solution:

Let S = {1, x, x², x³, x², ... x_n }.

Let α=a₀+a₁x+a₂x² + a₃t³…  +a_nxⁿ be any arbitrary number of P_n where a₀, a₁, a₂ …a_n ∈ F.

Then α is the linear combination of polynomials 1, x, x², x³... xⁿ over the field F.

S generates P_n(F).