Mixing
Weak Question: How to remember theorems and definitions?
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I find that I get a pretty good understanding of most theorems, but I have a really hard time remembering them to use when I'm doing proofs on quizzes. More specifically using the technical "equation" or "formulas" mentioned in the theorems.
For example I'm doing linear algebra right now in uni, and idk, one of the theorems is "A set of vectors $vec{v_1},...,vec{v_k}$ in $R^n$ is linearly dependent if and only if $vec{v_i} in Span[vec{v_1},...,vec{v_{i-1}}, vec{v_{I+1}},...,vec{v_k}]$ for some $i$, $1 leq i leq k$.
Now this is a really easy concept to get, and honestly this is one of the ones I can remember the best in the course but it was short so I could easily type it on here, but anyways, the idea is easy, but when I'm doing proofs I find it hard to rack my brain and search for this theorem that I have to use to prove, say a set is linearly dependent in this case (again, this is an easy example). I know most of the proofs, if I saw it I could use it easily, but when I'm writing a proof on a test, for some reason I just can't go "oh I can go to the next step using Theorem 1.3.2."
Any advice? I think it's all the complicated symbols and everything that's kind of clogging up my brain, it's easy to remember like "oh theorem that says what makes a set of vectors linearly dependent". But it's hard to go "Oh this vector in this question looks like it can be written using $Span[vec{v_1},...,vec{v_{i-1}}, vec{v_{I+1}},...,vec{v_k}]$". It's the use of the formal, like technical language part of the proof I have trouble with. Is my only option just purely memorizing the symbols and combinations of a,b,c, v, I, x, y, $in$, etc..
Thanks!
linear-algebra
asked Feb 2 at 23:55
mingming
4606
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add a comment |
$begingroup$
I find that I get a pretty good understanding of most theorems, but I have a really hard time remembering them to use when I'm doing proofs on quizzes. More specifically using the technical "equation" or "formulas" mentioned in the theorems.
For example I'm doing linear algebra right now in uni, and idk, one of the theorems is "A set of vectors $vec{v_1},...,vec{v_k}$ in $R^n$ is linearly dependent if and only if $vec{v_i} in Span[vec{v_1},...,vec{v_{i-1}}, vec{v_{I+1}},...,vec{v_k}]$ for some $i$, $1 leq i leq k$.
Now this is a really easy concept to get, and honestly this is one of the ones I can remember the best in the course but it was short so I could easily type it on here, but anyways, the idea is easy, but when I'm doing proofs I find it hard to rack my brain and search for this theorem that I have to use to prove, say a set is linearly dependent in this case (again, this is an easy example). I know most of the proofs, if I saw it I could use it easily, but when I'm writing a proof on a test, for some reason I just can't go "oh I can go to the next step using Theorem 1.3.2."
Any advice? I think it's all the complicated symbols and everything that's kind of clogging up my brain, it's easy to remember like "oh theorem that says what makes a set of vectors linearly dependent". But it's hard to go "Oh this vector in this question looks like it can be written using $Span[vec{v_1},...,vec{v_{i-1}}, vec{v_{I+1}},...,vec{v_k}]$". It's the use of the formal, like technical language part of the proof I have trouble with. Is my only option just purely memorizing the symbols and combinations of a,b,c, v, I, x, y, $in$, etc..
Thanks!
linear-algebra
asked Feb 2 at 23:55
mingming
4606
$endgroup$
$begingroup$
I suggest writing the theorems using just natural language without any symbols. There is no point in merely memorizing something unless it is important and it helps if you understand what it means (what it states).
$endgroup$
– Somos
Feb 3 at 1:24
$begingroup$
The thing is, most of the time I need like the "formal" definition, with equations and stuff to plug into the equations I'm given in the proof I'm trying to prove. Like when proving linear independence, I need to actually memorize the 0 = $c_1v_1 + … + c_nv_n$, and again that's an easy one but, I actually need to memorize that equation rather than, well "the only way that a set if linearly independent if the only solution to some linear combination of a set if all 0's". @Somos
$endgroup$
– ming
Feb 3 at 2:53
add a comment |
$begingroup$
I find that I get a pretty good understanding of most theorems, but I have a really hard time remembering them to use when I'm doing proofs on quizzes. More specifically using the technical "equation" or "formulas" mentioned in the theorems.
For example I'm doing linear algebra right now in uni, and idk, one of the theorems is "A set of vectors $vec{v_1},...,vec{v_k}$ in $R^n$ is linearly dependent if and only if $vec{v_i} in Span[vec{v_1},...,vec{v_{i-1}}, vec{v_{I+1}},...,vec{v_k}]$ for some $i$, $1 leq i leq k$.
Now this is a really easy concept to get, and honestly this is one of the ones I can remember the best in the course but it was short so I could easily type it on here, but anyways, the idea is easy, but when I'm doing proofs I find it hard to rack my brain and search for this theorem that I have to use to prove, say a set is linearly dependent in this case (again, this is an easy example). I know most of the proofs, if I saw it I could use it easily, but when I'm writing a proof on a test, for some reason I just can't go "oh I can go to the next step using Theorem 1.3.2."
Any advice? I think it's all the complicated symbols and everything that's kind of clogging up my brain, it's easy to remember like "oh theorem that says what makes a set of vectors linearly dependent". But it's hard to go "Oh this vector in this question looks like it can be written using $Span[vec{v_1},...,vec{v_{i-1}}, vec{v_{I+1}},...,vec{v_k}]$". It's the use of the formal, like technical language part of the proof I have trouble with. Is my only option just purely memorizing the symbols and combinations of a,b,c, v, I, x, y, $in$, etc..
Thanks!
linear-algebra
asked Feb 2 at 23:55
mingming
4606
$endgroup$
I find that I get a pretty good understanding of most theorems, but I have a really hard time remembering them to use when I'm doing proofs on quizzes. More specifically using the technical "equation" or "formulas" mentioned in the theorems.
For example I'm doing linear algebra right now in uni, and idk, one of the theorems is "A set of vectors $vec{v_1},...,vec{v_k}$ in $R^n$ is linearly dependent if and only if $vec{v_i} in Span[vec{v_1},...,vec{v_{i-1}}, vec{v_{I+1}},...,vec{v_k}]$ for some $i$, $1 leq i leq k$.
Now this is a really easy concept to get, and honestly this is one of the ones I can remember the best in the course but it was short so I could easily type it on here, but anyways, the idea is easy, but when I'm doing proofs I find it hard to rack my brain and search for this theorem that I have to use to prove, say a set is linearly dependent in this case (again, this is an easy example). I know most of the proofs, if I saw it I could use it easily, but when I'm writing a proof on a test, for some reason I just can't go "oh I can go to the next step using Theorem 1.3.2."
Any advice? I think it's all the complicated symbols and everything that's kind of clogging up my brain, it's easy to remember like "oh theorem that says what makes a set of vectors linearly dependent". But it's hard to go "Oh this vector in this question looks like it can be written using $Span[vec{v_1},...,vec{v_{i-1}}, vec{v_{I+1}},...,vec{v_k}]$". It's the use of the formal, like technical language part of the proof I have trouble with. Is my only option just purely memorizing the symbols and combinations of a,b,c, v, I, x, y, $in$, etc..
Thanks!
linear-algebra
linear-algebra
asked Feb 2 at 23:55
mingming
4606
asked Feb 2 at 23:55
mingming
4606
asked Feb 2 at 23:55
mingming
4606
asked Feb 2 at 23:55
mingming
4606
asked Feb 2 at 23:55
mingming
4606
4606
$begingroup$
I suggest writing the theorems using just natural language without any symbols. There is no point in merely memorizing something unless it is important and it helps if you understand what it means (what it states).
$endgroup$
– Somos
Feb 3 at 1:24
$begingroup$
The thing is, most of the time I need like the "formal" definition, with equations and stuff to plug into the equations I'm given in the proof I'm trying to prove. Like when proving linear independence, I need to actually memorize the 0 = $c_1v_1 + … + c_nv_n$, and again that's an easy one but, I actually need to memorize that equation rather than, well "the only way that a set if linearly independent if the only solution to some linear combination of a set if all 0's". @Somos
$endgroup$
– ming
Feb 3 at 2:53
add a comment |
$begingroup$
I suggest writing the theorems using just natural language without any symbols. There is no point in merely memorizing something unless it is important and it helps if you understand what it means (what it states).
$endgroup$
– Somos
Feb 3 at 1:24
$begingroup$
The thing is, most of the time I need like the "formal" definition, with equations and stuff to plug into the equations I'm given in the proof I'm trying to prove. Like when proving linear independence, I need to actually memorize the 0 = $c_1v_1 + … + c_nv_n$, and again that's an easy one but, I actually need to memorize that equation rather than, well "the only way that a set if linearly independent if the only solution to some linear combination of a set if all 0's". @Somos
$endgroup$
– ming
Feb 3 at 2:53
$begingroup$
I suggest writing the theorems using just natural language without any symbols. There is no point in merely memorizing something unless it is important and it helps if you understand what it means (what it states).
$endgroup$
– Somos
Feb 3 at 1:24
$begingroup$
I suggest writing the theorems using just natural language without any symbols. There is no point in merely memorizing something unless it is important and it helps if you understand what it means (what it states).
$endgroup$
– Somos
Feb 3 at 1:24
$begingroup$
The thing is, most of the time I need like the "formal" definition, with equations and stuff to plug into the equations I'm given in the proof I'm trying to prove. Like when proving linear independence, I need to actually memorize the 0 = $c_1v_1 + … + c_nv_n$, and again that's an easy one but, I actually need to memorize that equation rather than, well "the only way that a set if linearly independent if the only solution to some linear combination of a set if all 0's". @Somos
$endgroup$
– ming
Feb 3 at 2:53
$begingroup$
The thing is, most of the time I need like the "formal" definition, with equations and stuff to plug into the equations I'm given in the proof I'm trying to prove. Like when proving linear independence, I need to actually memorize the 0 = $c_1v_1 + … + c_nv_n$, and again that's an easy one but, I actually need to memorize that equation rather than, well "the only way that a set if linearly independent if the only solution to some linear combination of a set if all 0's". @Somos
$endgroup$
– ming
Feb 3 at 2:53
add a comment |
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$begingroup$
I suggest writing the theorems using just natural language without any symbols. There is no point in merely memorizing something unless it is important and it helps if you understand what it means (what it states).
$endgroup$
– Somos
Feb 3 at 1:24
$begingroup$
The thing is, most of the time I need like the "formal" definition, with equations and stuff to plug into the equations I'm given in the proof I'm trying to prove. Like when proving linear independence, I need to actually memorize the 0 = $c_1v_1 + … + c_nv_n$, and again that's an easy one but, I actually need to memorize that equation rather than, well "the only way that a set if linearly independent if the only solution to some linear combination of a set if all 0's". @Somos
$endgroup$
– ming
Feb 3 at 2:53