How can you prove that a linear transformation is injective? (a) What is another

Question

How can you prove that a linear transformation is injective? (a) What is another way you can prove that a linear transformation is injective? (b) Example: Prove that the linear transformation T : R2 P3(R) given by T (a, b) = ax2 b is injective. Use two different methods. How can you prove that a linear transformation is injective? (a) What is another way you can prove that a linear transformation is injective? (b) Example: Prove that the linear transformation T : R2 P3(R) given by T (a, b) = ax2 b is injective. Use two different methods. How can you prove that a linear transformation is injective? (a) What is another way you can prove that a linear transformation is injective? (b) Example: Prove that the linear transformation T : R2 P3(R) given by T (a, b) = ax2 b is injective. Use two different methods.

Explanation / Answer

If T: U--->V is inear transformation

then T is injective if

Method 1)- for x,y in U

if T(x)=T(y) then x=y

method 2) T:U-->V is linear transformation is injective if and only if

ker(T)={0}

To show that T:R2-->P3(R) define by T(a,b)=ax2-b is injective

By method 1

Let (a,b) and (c,d) in R2

consider

T(a,b)=T(c,d)

that is ax2-b=cx2-d

By comparing both side clearly

a=c and b=d

that is (a,b)=(c,d)

therfore T is injective

By method 2)

Befination of ker(T)

T:U--->V is inear transformation then kernal of Tg is define by

ker(T)={u inU / T(u)=0}

Here T:R2---->P3(R)

T(a,b)=ax2-b

ker(T)={(a,b) in R2/ T(a,b)=0}

={(a,b) in R2/ax2-b=0} is possible only if (a,b)=(0,0)

ker(T)={(0,0)}

therfore T is injective

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