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Linear Algebra

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Difficult

Two Zero Vectors

LINALG-OEUGZO

Let $V,W$ be vector spaces. Let $T:V\rightarrow W$ be a linear transformation. $\vec 0_V$ is the zero vector in $V$ and $\vec 0_W$ is the zero vector in $W$. Which of the following are correct?

Select ALL that apply.

A

Since $\vec 0_W$ is the zero vector in$W$, we know that:

$$\vec 0_W+T(\vec x)=T(\vec x)$$

...for all $\vec x\in V$. And since $\vec 0_V$ is the zero vector in $V$, for all:

$$\vec x\in V, \vec 0_V+\vec x=\vec x$$

Thus $T(\vec x)=T(\vec 0_V+\vec x)$.

Since $T$ is a linear transformation:

$$T(\vec x)=T(\vec 0_V+\vec x)=T(\vec 0_V)+T(\vec x)$$

Every vector space has a unique zero vector, so we see that $T(\vec 0_V)=\vec 0_W$.

B

Since $T$ is a linear transformation, for all $\vec x\in V$:

$$\vec 0_W=0T(\vec x)=T(0\vec x)=T(\vec 0_V)$$

This proves that $T(\vec 0_V)=\vec 0_W$.

C

It is not always true that $T(\vec 0_V)=\vec 0_W$.

D

It is always true that $T(\vec 0_V)=\vec 0_W$, but neither the argument in Choice 'A' nor the argument in Choice 'B' are correct.