Is every independent subset $S$ of a vector subspace $W$ such that $operatorname{card}(S)=dim(W)$ a basis for...
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I was reading a theorem that states that if $W$ is a subspace of a finite-dimensional vector space then every independent subset $S subset W$ can be extended to form a basis of $W$ by repeatedly adding vectors in way such that the resulting set continues to be linearly independent. But, can´t it be the case that we arrive by that process to a independent subset $S^*subset W$ with $operatorname{card}(S^*)=dim(W)$ that is not a basis of W?
I suspect that the answer is no, in that case,I would be grateful if you gave me some hints so I could write a proof
linear-algebra vector-spaces vectors
edited Nov 18 at 21:38
anomaly
17.1k42662
asked Nov 18 at 21:34
Victor Martini
61
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I was reading a theorem that states that if $W$ is a subspace of a finite-dimensional vector space then every independent subset $S subset W$ can be extended to form a basis of $W$ by repeatedly adding vectors in way such that the resulting set continues to be linearly independent. But, can´t it be the case that we arrive by that process to a independent subset $S^*subset W$ with $operatorname{card}(S^*)=dim(W)$ that is not a basis of W?
I suspect that the answer is no, in that case,I would be grateful if you gave me some hints so I could write a proof
linear-algebra vector-spaces vectors
edited Nov 18 at 21:38
anomaly
17.1k42662
asked Nov 18 at 21:34
Victor Martini
61
Hint: First prove that each basis in $W$ consists of exactly $n$ vectors, where $n=mbox{dim}(W)$. Thus every linearly independent subset of $n$ vectors forms a basis in $W$.
– Hasek
Nov 18 at 21:40
It might be useful to think of the sequence of extensions, $S = S_0 subsetneq S_1 subsetneq S_2 subsetneq cdots subsetneq S^*$, and their corresponding un-spanned subspaces, $W / langle S_0 rangle supsetneq W / langle S_1 rangle supsetneq cdots supsetneq W / langle S^* rangle$.
– Eric Towers
Nov 18 at 21:48
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I was reading a theorem that states that if $W$ is a subspace of a finite-dimensional vector space then every independent subset $S subset W$ can be extended to form a basis of $W$ by repeatedly adding vectors in way such that the resulting set continues to be linearly independent. But, can´t it be the case that we arrive by that process to a independent subset $S^*subset W$ with $operatorname{card}(S^*)=dim(W)$ that is not a basis of W?
I suspect that the answer is no, in that case,I would be grateful if you gave me some hints so I could write a proof
linear-algebra vector-spaces vectors
edited Nov 18 at 21:38
anomaly
17.1k42662
asked Nov 18 at 21:34
Victor Martini
61
I was reading a theorem that states that if $W$ is a subspace of a finite-dimensional vector space then every independent subset $S subset W$ can be extended to form a basis of $W$ by repeatedly adding vectors in way such that the resulting set continues to be linearly independent. But, can´t it be the case that we arrive by that process to a independent subset $S^*subset W$ with $operatorname{card}(S^*)=dim(W)$ that is not a basis of W?
I suspect that the answer is no, in that case,I would be grateful if you gave me some hints so I could write a proof
linear-algebra vector-spaces vectors
linear-algebra vector-spaces vectors
edited Nov 18 at 21:38
anomaly
17.1k42662
asked Nov 18 at 21:34
Victor Martini
61
edited Nov 18 at 21:38
anomaly
17.1k42662
asked Nov 18 at 21:34
Victor Martini
61
edited Nov 18 at 21:38
anomaly
17.1k42662
edited Nov 18 at 21:38
anomaly
17.1k42662
edited Nov 18 at 21:38
anomaly
17.1k42662
17.1k42662
asked Nov 18 at 21:34
Victor Martini
61
asked Nov 18 at 21:34
Victor Martini
61
asked Nov 18 at 21:34
Victor Martini
61
61
Hint: First prove that each basis in $W$ consists of exactly $n$ vectors, where $n=mbox{dim}(W)$. Thus every linearly independent subset of $n$ vectors forms a basis in $W$.
– Hasek
Nov 18 at 21:40
It might be useful to think of the sequence of extensions, $S = S_0 subsetneq S_1 subsetneq S_2 subsetneq cdots subsetneq S^*$, and their corresponding un-spanned subspaces, $W / langle S_0 rangle supsetneq W / langle S_1 rangle supsetneq cdots supsetneq W / langle S^* rangle$.
– Eric Towers
Nov 18 at 21:48
add a comment |
Hint: First prove that each basis in $W$ consists of exactly $n$ vectors, where $n=mbox{dim}(W)$. Thus every linearly independent subset of $n$ vectors forms a basis in $W$.
– Hasek
Nov 18 at 21:40
It might be useful to think of the sequence of extensions, $S = S_0 subsetneq S_1 subsetneq S_2 subsetneq cdots subsetneq S^*$, and their corresponding un-spanned subspaces, $W / langle S_0 rangle supsetneq W / langle S_1 rangle supsetneq cdots supsetneq W / langle S^* rangle$.
– Eric Towers
Nov 18 at 21:48
Hint: First prove that each basis in $W$ consists of exactly $n$ vectors, where $n=mbox{dim}(W)$. Thus every linearly independent subset of $n$ vectors forms a basis in $W$.
– Hasek
Nov 18 at 21:40
Hint: First prove that each basis in $W$ consists of exactly $n$ vectors, where $n=mbox{dim}(W)$. Thus every linearly independent subset of $n$ vectors forms a basis in $W$.
– Hasek
Nov 18 at 21:40
It might be useful to think of the sequence of extensions, $S = S_0 subsetneq S_1 subsetneq S_2 subsetneq cdots subsetneq S^*$, and their corresponding un-spanned subspaces, $W / langle S_0 rangle supsetneq W / langle S_1 rangle supsetneq cdots supsetneq W / langle S^* rangle$.
– Eric Towers
Nov 18 at 21:48
It might be useful to think of the sequence of extensions, $S = S_0 subsetneq S_1 subsetneq S_2 subsetneq cdots subsetneq S^*$, and their corresponding un-spanned subspaces, $W / langle S_0 rangle supsetneq W / langle S_1 rangle supsetneq cdots supsetneq W / langle S^* rangle$.
– Eric Towers
Nov 18 at 21:48
add a comment |
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Hint: First prove that each basis in $W$ consists of exactly $n$ vectors, where $n=mbox{dim}(W)$. Thus every linearly independent subset of $n$ vectors forms a basis in $W$.
– Hasek
Nov 18 at 21:40
It might be useful to think of the sequence of extensions, $S = S_0 subsetneq S_1 subsetneq S_2 subsetneq cdots subsetneq S^*$, and their corresponding un-spanned subspaces, $W / langle S_0 rangle supsetneq W / langle S_1 rangle supsetneq cdots supsetneq W / langle S^* rangle$.
– Eric Towers
Nov 18 at 21:48