Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Let $T$ be the linear transformation represented by the matrix

$$ \left( \begin{array}{ccc} 1 & 1 & 0 & 3 \\ 1 & 1 & 1 & 5 \\ 2 & 2 & 1 & 8 \end{array} \right) $$

One can easily calculate that the image of this as a map from $\mathbb{R}^4\to\mathbb{R}^3$ is 3.

Call the above matrix $A$. Now consider the space $V$ of linear maps $B$ from $\mathbb{R}^2\to\mathbb{R}^4$ satisfying $AB=0$. Plainly $B$ is in $V$ iff the image of $B$ is in the kernel of $T$.

What is the dimension of $V$? Directly calculating, I get 4. However, it seems that one could argue that the dimension is the dimension of the kernel of $T$, which is $2$. What is the flaw in reasoning?

share|improve this question

2 Answers 2

up vote 1 down vote accepted

To rephrase you slightly, $V$ is simply the space of linear maps from $\mathbb{R}^2 \rightarrow ker(T)$. Note that $ker(T)$ is 2 dimensional, as you've said, and then linear maps from a 2 dimensional space to a 2 dimensional space form a vector space of $2 \cdot 2 = 4$ dimensions.

It is not simply the dimension of $ker(T)$ because you've got to account for the dimension of the domain as well.

share|improve this answer

If the rank of $A$ is $3$, as you say, then it's kernel has dimension $4-3 = 1$, by the rank-nullity-theorem. But the rank is two (the third row is the sum of the first 2 and hence the kernel has dimension 2). Now let's $V$ is, as you say, the space of all linear maps $\mathbb R^2\to \mathbb R^4$, whose image is contained in $\ker T$. So we are asked for the dimension of $L(\mathbb R^2, \ker T)$, that is the space of linear maps from $\mathbb R^2$ to another two dimensional space. The dimension of $L(X,Y)$ is known to be $\dim X \cdot \dim Y$ (if you represent them as matrices, a basis is easily given as those matrices which have exactly one 1 and otherwise zeros), so $\dim V = 2 \cdot 2 = 4$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.