Son Gohan
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My friend sent me these from Canada . Apparently these are undergrad level vectors and I do not know how to solve these. I need someone to help me out as to what are the answers to these
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Cryptokyo
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(Original post by Son Gohan)
My friend sent me these from Canada . Apparently these are undergrad level vectors and I do not know how to solve these. I need someone to help me out as to what are the answers to these
Have you learnt the material relevant for this (i.e. vector spaces, vector subspaces, spanning sets, linear dependence etc)? Because otherwise there is not much point in doing these problems yet. But if you still want to have help with solving them, I am more than happy to help.
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Son Gohan
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(Original post by Cryptokyo)
Have you learnt the material relevant for this (i.e. vector spaces, vector subspaces, spanning sets, linear dependence etc)? Because otherwise there is not much point in doing these problems yet. But if you still want to have help with solving them, I am more than happy to help.
I am still in A levels, so hopefully I ll learn them when I get to uni this year ! However, my friend has no one to ask help from, so she wanted to know the answers and asked me if I could gather them from somewhere.I will be really glad if you could help me out !
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Cryptokyo
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Son Gohan

I am hesitant to give the actual true or false answers but I shall give some incomplete working that should help your friend reach the correct answer. Of course, you're friend may have got some of them right but this should help understanding and reaffirm that they got a question right if the logic agrees with them.

Regarding attachment one.

1. w\in\text{span}\lbrace v_{1},v_{2},v_{3}\rbrace \Rightarrow w=\lambda_{1}v_{1}+\lambda_{2}v_{2}+\lambda_{3}v_{3} for some constants \lambda_{1},\lambda_{2},\lambda_{3}. Do we require at least two of \lambda_{1},\lambda_{2},\lambda_{3} to be zero for w to be in the vector space?

2. Consider the subset \left\lbrace \begin{pmatrix}0\\0\end{pmatrix},\begin{pmatrix}1\\1 \end{pmatrix}\right\rbrace \subset \mathbb{R}^{2}. Does this have closure under scalar multiplication and addition?

3. All vectors in our solution set must be of the same dimension as the vector x, which is of degree n from the n unknowns. What does this imply?

4. Let V=\text{span}\left\lbrace v_{1},v_{2},\ldots,v_{n} \right \rbrace, then if x,y\in V then x=\lambda_{1}v_{1}+\lambda_{2}v_{2}+ \cdots+\lambda_{n}v_{n} and y=\mu_{1}v_{1}+\mu_{2}v_{2}+ \cdots+\mu_{n}v_{n} where \lambda_{i} and \mu_{i} are some real constants. Using this, is it true that x+y\in V and ax\in V for some constant a?

5. Consider the non-zero vector x\in\text{span}\lbrace u \rbrace=\text{span}\lbrace v\rbrace. Then x=\lambda u=\mu v for some non-zero constants \lambda and \mu. What can we conclude from this?
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Cryptokyo
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Regarding attachment two. Note I have not bothered to bold vectors in either set of questions.

1. v_{1},v_{2},v_{3},v_{4} are linearly independent if and only if \lambda_{1}v_{1}+\lambda_{2}v_{2}+\lambda_{3}v_{3}+\lambda_{4}v_{4}=0 implies that \lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}=0. Now if we fix \lambda_{4}=0, do we meet the condition for v_{1},v_{2},v_{3} to be linearly independent?

2. If v_{1},v_{2},v_{3} are linearly dependent, then we have a solution to \lambda_{1}v_{1}+\lambda_{2}v_{2}+\lambda_{3}v_{3}=0 where at least one of the constants \lambda_{1},\lambda_{2},\lambda_{3} is non-zero. W.l.o.g. Suppose that \lambda_{1}\neq 0. Then v_{1}=-\frac{\lambda_{2}}{\lambda_{1}}v_{2}-\frac{\lambda_{3}}{\lambda_{1}}v_{3}. Is it necessarily true that at least one of \lambda_{2},\lambda_{3} is zero.

3. Consider the linear dependent set of vectors \left\lbrace \begin{pmatrix}1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix}0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix}0 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix}1 \\ 1 \\ -1 \end{pmatrix}\right\rbrace. If we remove  \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}, do we still have a set of linearly dependent vectors?

4. Consider the linearly dependent set of vectors given in the question 3 example, do they span \mathbb{R}^{3}?

5. Consider S=\lbrace v_{1},v_{2},\ldots, v_{n}\rbrace. Consider the equation \lambda_{1}v_{1}+\lambda_{2}v_{2}+\cdots +\lambda_{n}v_{n}=0. If v_{n} is the zero vector then what does this mean for \lambda_{n}? Does this satisfy our condition for linear independence as stated in the hint for question 1.

6. Consider the linearly independent set of vectors \left\lbrace \begin{pmatrix}1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix}0 \\ 1 \\ 0 \end{pmatrix} \right\rbrace. Do they span \mathbb{R}^{3}?

Son Gohan I am happy to clarify anything further but it is better if I refrain from directly telling the answers.
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Son Gohan
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(Original post by Cryptokyo)
Regarding attachment two. Note I have not bothered to bold vectors in either set of questions.

1. v_{1},v_{2},v_{3},v_{4} are linearly independent if and only if \lambda_{1}v_{1}+\lambda_{2}v_{2}+\lambda_{3}v_{3}+\lambda_{4}v_{4}=0 implies that \lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}=0. Now if we fix \lambda_{4}=0, do we meet the condition for v_{1},v_{2},v_{3} to be linearly independent?

2. If v_{1},v_{2},v_{3} are linearly dependent, then we have a solution to \lambda_{1}v_{1}+\lambda_{2}v_{2}+\lambda_{3}v_{3}=0 where at least one of the constants \lambda_{1},\lambda_{2},\lambda_{3} is non-zero. W.l.o.g. Suppose that \lambda_{1}\neq 0. Then v_{1}=-\frac{\lambda_{2}}{\lambda_{1}}v_{2}-\frac{\lambda_{3}}{\lambda_{1}}v_{3}. Is it necessarily true that at least one of \lambda_{2},\lambda_{3} is zero.

3. Consider the linear dependent set of vectors \left\lbrace \begin{pmatrix}1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix}0 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix}0 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix}1 \\ 1 \\ -1 \end{pmatrix}\right\rbrace. If we remove  \begin{pmatrix} 1 \\ 1 \\ -1 \end{pmatrix}, do we still have a set of linearly dependent vectors?

4. Consider the linearly dependent set of vectors given in the question 3 example, do they span \mathbb{R}^{3}?

5. Consider S=\lbrace v_{1},v_{2},\ldots, v_{n}\rbrace. Consider the equation \lambda_{1}v_{1}+\lambda_{2}v_{2}+\cdots +\lambda_{n}v_{n}=0. If v_{n} is the zero vector then what does this mean for \lambda_{n}? Does this satisfy our condition for linear independence as stated in the hint for question 1.

6. Consider the linearly independent set of vectors \left\lbrace \begin{pmatrix}1 \\ 0 \\ 0 \end{pmatrix}, \begin{pmatrix}0 \\ 1 \\ 0 \end{pmatrix} \right\rbrace. Do they span \mathbb{R}^{3}?

Son Gohan I am happy to clarify anything further but it is better if I refrain from directly telling the answers.
Thanks a lot ! I ve messaged it to her .
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Son Gohan
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Hi She sent me this
F, F, T, T, T T, F, F, F, T, F
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