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Please help me to find this problem

Let $W$ be a vector space over R and let $T:R^6→W$ be a linear transformation such that $S=\{T(0,1,0,0,0,0),\, T(0,0,0,1,0,0),\, T(0,0,0,0,0,1)\}$ spans W. Which one of the following must be true?

a) $S$ is a basis of $W$

b) $T(R^6)≠W$

c)$\{T(1,0,0,0,0,0),\, T(0,0,1,0,0,0),\, T(0,0,0,0,1,0)\}$ spans $W$

d) $\ker(T)$ contains more than one element

Do you have a tip for me to solve this ? Thank you in advance!

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    Tip1: For a basis of W, how many elements do you need? – 2017-01-24

1 Answers 1

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$$\dim \mathbb{R}^6=\dim \ker T+\dim\text{Im }T\Rightarrow 6=\dim \ker T+\underbrace{\dim \text{Im }T}_{\le 3}\Rightarrow \dim \ker T\ge 3$$

$$\Rightarrow \ker T\ne\{0\}\Rightarrow \ker T\text{ contains more than one element.}$$

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    I can't understand the part that you have said " dim ImT less than equals to 3". @Fernando Revilla – 2017-05-15