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Proving a set is a linear subspace of a vector space
vector space and its subspaceHow to find equation system describing affine space, having base of linear space and a vectorIf a vector space has a proper subspace isomorphic to itself, it has infinite dimension?What exactly is a linear space?Linear algebra too early.Proof that every subspace is a vector spaceHow to prove a type of functions is a subspace of the vector space of all functions.Vector space generated by set intersectionProving a subset is a subspace to a Vector space.Linear Algebra, proving subset is a subspace
$begingroup$
Suppose $E$ is a vector space over $K$ and $F$ is a subset of $E$.
For $F$ to be a linear subspace then $forall u,v in F$ and $forall alpha,betain K$, $alpha u + beta v in F$
My professor has been using $beta=1$, how is that okay?
linear-algebra vector-spaces
asked 15 hours ago
LuywLuyw
275
$endgroup$
add a comment |
$begingroup$
Suppose $E$ is a vector space over $K$ and $F$ is a subset of $E$.
For $F$ to be a linear subspace then $forall u,v in F$ and $forall alpha,betain K$, $alpha u + beta v in F$
My professor has been using $beta=1$, how is that okay?
linear-algebra vector-spaces
asked 15 hours ago
LuywLuyw
275
$endgroup$
add a comment |
$begingroup$
Suppose $E$ is a vector space over $K$ and $F$ is a subset of $E$.
For $F$ to be a linear subspace then $forall u,v in F$ and $forall alpha,betain K$, $alpha u + beta v in F$
My professor has been using $beta=1$, how is that okay?
linear-algebra vector-spaces
asked 15 hours ago
LuywLuyw
275
$endgroup$
Suppose $E$ is a vector space over $K$ and $F$ is a subset of $E$.
For $F$ to be a linear subspace then $forall u,v in F$ and $forall alpha,betain K$, $alpha u + beta v in F$
My professor has been using $beta=1$, how is that okay?
linear-algebra vector-spaces
linear-algebra vector-spaces
asked 15 hours ago
LuywLuyw
275
asked 15 hours ago
LuywLuyw
275
edited 14 hours ago
Luyw
asked 15 hours ago
LuywLuyw
275
asked 15 hours ago
LuywLuyw
275
asked 15 hours ago
LuywLuyw
275
275
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
If $beta=0$, then $alpha u + beta v = alpha u in F$ which should be true if $F$ is a subspace of the vector space $E$
So, you can suppose that $beta neq 0$, now divide things by $beta$ which is non-zero. Since the statement must be true for all $alpha$, you can assume that $alpha/beta$ is another $alphain mathbbR$.
answered 14 hours ago
stressed outstressed out
6,5431939
$endgroup$
$begingroup$
I see, thank you!
$endgroup$
– Luyw
14 hours ago
$begingroup$
@Luyw You're welcome.
$endgroup$
– stressed out
14 hours ago
add a comment |
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
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active
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votes
$begingroup$
If $beta=0$, then $alpha u + beta v = alpha u in F$ which should be true if $F$ is a subspace of the vector space $E$
So, you can suppose that $beta neq 0$, now divide things by $beta$ which is non-zero. Since the statement must be true for all $alpha$, you can assume that $alpha/beta$ is another $alphain mathbbR$.
answered 14 hours ago
stressed outstressed out
6,5431939
$endgroup$
$begingroup$
I see, thank you!
$endgroup$
– Luyw
14 hours ago
$begingroup$
@Luyw You're welcome.
$endgroup$
– stressed out
14 hours ago
add a comment |
$begingroup$
If $beta=0$, then $alpha u + beta v = alpha u in F$ which should be true if $F$ is a subspace of the vector space $E$
So, you can suppose that $beta neq 0$, now divide things by $beta$ which is non-zero. Since the statement must be true for all $alpha$, you can assume that $alpha/beta$ is another $alphain mathbbR$.
answered 14 hours ago
stressed outstressed out
6,5431939
$endgroup$
$begingroup$
I see, thank you!
$endgroup$
– Luyw
14 hours ago
$begingroup$
@Luyw You're welcome.
$endgroup$
– stressed out
14 hours ago
add a comment |
$begingroup$
If $beta=0$, then $alpha u + beta v = alpha u in F$ which should be true if $F$ is a subspace of the vector space $E$
So, you can suppose that $beta neq 0$, now divide things by $beta$ which is non-zero. Since the statement must be true for all $alpha$, you can assume that $alpha/beta$ is another $alphain mathbbR$.
answered 14 hours ago
stressed outstressed out
6,5431939
$endgroup$
If $beta=0$, then $alpha u + beta v = alpha u in F$ which should be true if $F$ is a subspace of the vector space $E$
So, you can suppose that $beta neq 0$, now divide things by $beta$ which is non-zero. Since the statement must be true for all $alpha$, you can assume that $alpha/beta$ is another $alphain mathbbR$.
answered 14 hours ago
stressed outstressed out
6,5431939
edited 14 hours ago
answered 14 hours ago
stressed outstressed out
6,5431939
answered 14 hours ago
stressed outstressed out
6,5431939
answered 14 hours ago
stressed outstressed out
6,5431939
6,5431939
$begingroup$
I see, thank you!
$endgroup$
– Luyw
14 hours ago
$begingroup$
@Luyw You're welcome.
$endgroup$
– stressed out
14 hours ago
add a comment |
$begingroup$
I see, thank you!
$endgroup$
– Luyw
14 hours ago
$begingroup$
@Luyw You're welcome.
$endgroup$
– stressed out
14 hours ago
$begingroup$
I see, thank you!
$endgroup$
– Luyw
14 hours ago
$begingroup$
I see, thank you!
$endgroup$
– Luyw
14 hours ago
$begingroup$
@Luyw You're welcome.
$endgroup$
– stressed out
14 hours ago
$begingroup$
@Luyw You're welcome.
$endgroup$
– stressed out
14 hours ago
add a comment |
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