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 Dec26 comment If two sets (including A) are mutually exclusive, then A is the union of the sets. Thanks! I was looking at it in the wrong "direction"! Aug15 comment A basic doubt on linear dependence of vectors I think you mean "since t is not in the subspace spanned by u, v and w." Dec7 comment Why is $\mathbf{\hat{x}} = \sum_{j=2}^n (\mathbf{x}\cdot\mathbf{u}_j)\mathbf{u}_j = \mathbf{x} - (\mathbf{x}\cdot \mathbf{u})\mathbf{u}$? Thanks, David! That's what I needed! Weird how it's so obvious after someone told you where to look... Oct20 comment Different results for row reduction in Matlab @J.M. I am using MATLAB 7.11.0 (R2010b). Oct20 comment Different results for row reduction in Matlab @J.M. and Agusti, I think I might have narrowed it down to a rounding error, causing the top and bottom row to be slightly different (but not shown due to lack of decimals shown in Matlab). As you are both more experienced here than me, do you think I should answer my own post or delete it? Thank you! Oct10 comment Showing that $\{ 1, \cos t, \cos^2 t, \dots, \cos^6 t \}$ is a linearly independent set Thank you for your answer, Arturo! I think my problem might stem from a misunderstanding of what linear independence requires. I thought for functions to be linearly independent, they need to be so for all $\mathbb{R}$; however, it seems like it's the other way around: i.e. if a set of functions are not linearly dependent for all $\mathbb{R}$, then they are linearly independent. So if we can show one example of linear independence, then we're done. Is that the gist of what you were saying? Oct10 comment Showing that $\{ 1, \cos t, \cos^2 t, \dots, \cos^6 t \}$ is a linearly independent set Thank you for your answer, André! Forgive me for asking, but it is not obvious to me why the fact that $\cos t$ can take on more than 6 different values makes a contradiction? Oct8 comment Showing that a one-to-one linear transformation maps a linearly independent set onto a linearly independent set Thank you for the counterexample! Sep4 comment Showing that $\mathbf{u}=\mathbf{0}$ for $c\mathbf{u}=\mathbf{0}$, $c$ nonzero Good point, thanks! Sep4 comment Showing that $\mathbf{u}=\mathbf{0}$ for $c\mathbf{u}=\mathbf{0}$, $c$ nonzero @Srivatsan: Good to know, thanks! Sep4 comment Showing that $\mathbf{u}=\mathbf{0}$ for $c\mathbf{u}=\mathbf{0}$, $c$ nonzero @Chandrasekhar: Thanks! Sep1 comment editing signal in frequency domain and converting back to time domain You might want to post this at StackOverflow as well, for more answers (or even avp.stackexchange.com under signal-processing). Aug31 comment Meaning of, and how to verify, a vector space *over* $\mathbb{R}$ +1 Thank you for the detailed explanation, Michael! Aug31 comment Meaning of, and how to verify, a vector space *over* $\mathbb{R}$ @D B Lim: I put good advice into action!:) Aug31 comment Meaning of, and how to verify, a vector space *over* $\mathbb{R}$ @Willie: Thank you, that explained it! I suggest you consider writing that textbook! gary: I have not seen or learnt about fields or finite fields yet. I'll keep an eye out for that though! Aug31 comment Help with the proof of the characterization of linearly dependent sets @D B Lim: Thank you D B Lim! You're right, Lay has given me the impression that linear algebra is mostly about matrices... I'll pick the book up at my library right away! Thank you for the helpful suggestion! Aug31 comment Help with the proof of the characterization of linearly dependent sets Thanks D B Lim! +1! Vector spaces are the next chapter in the book, so I'll have a go at these once I've had a look in that chapter. Aug30 comment Help with the proof of the characterization of linearly dependent sets @D B Lim: Thank you! That makes perfect sense! Aug30 comment Help with the proof of the characterization of linearly dependent sets Thank you! I think I got it! Now the point with the second part, where v_1 is not zero, is that then not all numbers c_2 -> c_p can be zero (for it to be linearly dependent), which in turn leads to the fact that v_j is a linear combination of the preceding vectors... Have I kind of got it? I have at least got what I initially asked for, so I'm marking your post as the answer:) Aug30 comment Help with the proof of the characterization of linearly dependent sets Sorry, I didn't mean to rephrase it. Do you mean that we're simply saying that v_1 is a trivial linear combination in this case; and that it could just as fine be a non-trivial combination as well (but for the sake of the proof we're assuming the former)?