1. ## Dimension of Vectorspaces

Hi

how to show that the dimension of a subvector space ist smaller or at least equal to the dimension of the main vectorspace?

I know that the dimension is the length of the basis but how to use this information here?

thank you
greetings

2. Originally Posted by Herbststurm
Hi

how to show that the dimension of a subvector space ist smaller or at least equal to the dimension of the main vectorspace?

I know that the dimension is the length of the basis but how to use this information here?

thank you
greetings
If $V$ is a subspace of $W$ then say $\text{dim}(V) > \text{dim}(W)$ then it means we can find a basis set for $V$, $\{v_1,...,v_n\}$. But these vectors are linearly independent. The problem is that a set of linearly independent vectors cannot exceede the dimension and we see that $v_1,...,v_n$ exceede the dimension of $W$. This is a contradiction.

3. Sorry, I don't understand your post

How could you write that the dimension of the subspace is bigger than the main space?

Ad why do you say the base is linear dependent. I thought a base is always a linear indipendent creation system?

greetings

4. Originally Posted by Herbststurm
Sorry, I don't understand your post

How could you write that the dimension of the subspace is bigger than the main space?

Ad why do you say the base is linear dependent. I thought a base is always a linear indipendent creation system?

greetings
Say $\text{dim}(W) = 3$ and $V\subseteq W$ as a subspace has $\text{dim}(V) = 4$. Then this means $V$ has a basis of four elements, say, $\{ v_1,v_2,v_3,v_4\}$. This is a linearly independent set in $V$. Since $v_i \in V \implies v_i \in W$ because $V\subseteq W$. Therefore, $\{ v_1,v_2,v_3,v_4\}$ is a linearly independent set in $W$. But it is impossible to have a linearly independent set so that the number of elements in the set exceede the dimension of the space. Here the number of elements are $4$ and the dimension of W is $3$ and we see that $4>3$. This is a contradiction. Thus, the basis for $V$ must have fewer or equal to than $3$ elements.