[SOLVED] Onto linear transfomation

dwsmith

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Let V be the real vector space of all real 2x3 matrices, and let W be the real vector space of all real 4x1 column vectors. If T is a linear transformation from V onto W, what is the dimension of the subspace {v \(\displaystyle \in\) V: T(v) = 0}?

ker(v)=nullity; therefore, the range of v=6.

\(\displaystyle \begin{bmatrix}
a & b & c\\
d & e & f
\end{bmatrix}\)

I know the answer is 2 but how do I show it?
 

Bruno J.

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\(\displaystyle 6 = \mbox{dim }V = \mbox{dim } T(V) + \mbox{dim ker }V = 4 + \mbox{dim ker }V\)
 
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dwsmith

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\(\displaystyle 6 = \mbox{dim }V = \mbox{dim } T(V) + \mbox{dim ker }V = 4 + \mbox{dim ker }V\)
Then that leaves us with:

6 = dim V = dim T(v) + 0 = 4 + 0

6 = dim V = dim T(v) = 4

Will this equality always work for any transformation?
 

Defunkt

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Then that leaves us with:

6 = dim V = dim T(v) + 0 = 4 + 0

6 = dim V = dim T(v) = 4

Will this equality always work for any transformation?
6=4?


\(\displaystyle Ker(T) = \{v\in V : T(v) = 0\}\). You want to find \(\displaystyle dim ~ Ker(T)\). By a theorem (it is rather easy to prove), as Bruno had mentioned, we have that \(\displaystyle dim ~ V = dim ~ Im(T) + dim ~ Ker(T)\). Since T is onto a space of dimension 4, we get that \(\displaystyle dim ~ Im(T) = 4\). However, \(\displaystyle dim ~ V = 6\) and so: \(\displaystyle dim ~ V = dim ~ Im(T) + dim ~ Ker(T) \Rightarrow 6 = 4 + dim ~ Ker(T) \) \(\displaystyle \Rightarrow dim ~ Ker(T) = 6-4 = 2\)
 

dwsmith

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6=4?


\(\displaystyle Ker(T) = \{v\in V : T(v) = 0\}\). You want to find \(\displaystyle dim ~ Ker(T)\). By a theorem (it is rather easy to prove), as Bruno had mentioned, we have that \(\displaystyle dim ~ V = dim ~ Im(T) + dim ~ Ker(T)\). Since T is onto a space of dimension 4, we get that \(\displaystyle dim ~ Im(T) = 4\). However, \(\displaystyle dim ~ V = 6\) and so: \(\displaystyle dim ~ V = dim ~ Im(T) + dim ~ Ker(T) \Rightarrow 6 = 4 + dim ~ Ker(T) \) \(\displaystyle \Rightarrow dim ~ Ker(T) = 6-4 = 2\)

6= dim V

6 = dim T(v) = 4.... I don't know how you came up with 6=4 from that because 6-4=2 = dim T(v) = 4-4=0