Linear Transformation

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. . Let T₁ and T₂ be linear operators on R² defined by \Gamma_{1}(x,y)=(y,x) and T_{2}(x,y)=(x,0)

Compute T_{1}+T_{2},T_{2}T_{1},T_{1}T_{2},{T_{1}}^{2}T_{2} As~L(R^{\angle}) R2 is a vector space, which is closed under addion and scalar

A\rightarrow(B

Sol multiplication

T_{1}+T_{2},T_{2}T_{1},T_{1}T_{2},{T_{1}}^{2} {}^{2},{T_{2}}^{2} 1. (T_{1}+T_{2})(x,y) = T (x, y) + T₂ (x, y) for (x, y) ER² =(y,x)+(x,0) = (y+x, x+0) =(x+y,x)

09(B)

xy ∈ Reals [y+x=x+y」

2. (T₂ T₁) (x, y) = T2 [T₁ (x, y)] =T_{2}[y,x] =(y, 0)

3. (T₁ T2) (x, y) =T_{1}[T_{2}(x,y)] =T_{1}[x,0] =(0,x)

4. ({T_{1}}^{2})(x,y)=T_{1}[T_{1}(x,y)] =T_{1}(y,x) =(x,y)=I[(x,y)] T_{1}^{2}=I where I is an identity operator

[by def. of T₁]

[by def, of T2]

[by def. of T

C.

[by def. of T

5. ({T_{2}}^{2})(x,y)=T_{2}[T_{2}(x,y)] =T_{2}[x,0] =(x,0).=T_{2}(x,y)

f\cdot{T_{2}}^{2}

ample 4. (a) Let T_{1}:R^{3}\rightarrow R^{2}; and T_{2}:R^{3}\rightarrow R^{2} be two

Compute T_{1}+T_{2},T_{2}T_{1},T_{1}T_{2},{T_{1}}^{2}T_{2} As~L(R^{\angle}) R2 is a vector space, which is closed under addion and scalar

A\rightarrow(B

Sol multiplication

T_{1}+T_{2},T_{2}T_{1},T_{1}T_{2},{T_{1}}^{2} {}^{2},{T_{2}}^{2} 1. (T_{1}+T_{2})(x,y) = T (x, y) + T₂ (x, y) for (x, y) ER² =(y,x)+(x,0) = (y+x, x+0) =(x+y,x)

09(B)

xy ∈ Reals [y+x=x+y」

2. (T₂ T₁) (x, y) = T2 [T₁ (x, y)] =T_{2}[y,x] =(y, 0)

3. (T₁ T2) (x, y) =T_{1}[T_{2}(x,y)] =T_{1}[x,0] =(0,x)

4. ({T_{1}}^{2})(x,y)=T_{1}[T_{1}(x,y)] =T_{1}(y,x) =(x,y)=I[(x,y)] T_{1}^{2}=I where I is an identity operator

[by def. of T₁]

[by def, of T2]

[by def. of T

C.

[by def. of T

5. ({T_{2}}^{2})(x,y)=T_{2}[T_{2}(x,y)] =T_{2}[x,0] =(x,0).=T_{2}(x,y)

f\cdot{T_{2}}^{2}

ample 4. (a) Let T_{1}:R^{3}\rightarrow R^{2}; and T_{2}:R^{3}\rightarrow R^{2} be two