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 Dec 12 comment A context-free grammar for the language $L = \{ a^ib^jc^k \space|\space 0 \leq i \leq j \leq i + k \}$ Ok, thank you very much! Dec 12 comment A context-free grammar for the language $L = \{ a^ib^jc^k \space|\space 0 \leq i \leq j \leq i + k \}$ How about the third attempt? Dec 9 comment Row Reduction with Cofactor Expansion Oh, I see. Thank you. Dec 9 comment Row Reduction with Cofactor Expansion Thank you! So, why does my calculator and wolframalpha say 396? wolframalpha.com/input/… Nov 19 comment Describe the smallest subspace of $M_{2\times 2}$ that contains matrices… Ah. In that case, the set IS the smallest subspace. It is the basis. Nov 19 comment Describe the smallest subspace of $M_{2\times 2}$ that contains matrices… @BrianM.Scott, how does this translate to a basis? I'm used to row reducing into free variables that I translate into a general solution. Nov 19 comment Describe the smallest subspace of $M_{2\times 2}$ that contains matrices… I guess my first instinct is to think, "No solution!". So if it's fine, then how do I translate that to a basis? Is it a free variable column, and I just disregard the column of 0s before it? Nov 19 comment Describe the smallest subspace of $M_{2\times 2}$ that contains matrices… I actually did that, though I got [1 0 0 2; 0 1 0 -4; 0 0 0 -4]. I'm not sure what to make of the last row. Do I just disregard it? Nov 7 comment How can I prove a basis (for RowA, NulA, ColA, or NulA^T) with orthogonality conditions? The edition is the 4th Nov 7 comment How can I prove a basis (for RowA, NulA, ColA, or NulA^T) with orthogonality conditions? Linear Algebra by David C Lay, Chapter 4 Nov 7 comment How can I prove a basis (for RowA, NulA, ColA, or NulA^T) with orthogonality conditions? Ok. I'll have to figure how to extract the SVD from A. Nov 7 comment Let $B =${$1-t^2, t-t^2, 2-2t+t^2$}. Check that $B$ is a basis for $P_2$ and find $[3+t-6t^2]_B$ Okay. So, how would I find this from B? How would I prove that B is the basis for P_2? I'm not sure where to start. Nov 7 comment How can I prove a basis (for RowA, NulA, ColA, or NulA^T) with orthogonality conditions? @LittleO, how so? Nov 7 comment Find a basis for the set of vectors in $\mathbb{R}^4$ in the subspace (hyperplane) $x_1 +x_2 + 2x_3 + x_4 = 0, x_1 + 2x_2-x_3=0$ A sheet my professor gave out as practice problems for our test. Nov 7 comment Find a basis for the set of vectors in $\mathbb{R}^4$ in the subspace (hyperplane) $x_1 +x_2 + 2x_3 + x_4 = 0, x_1 + 2x_2-x_3=0$ The actual problem itself says that in the question. Nov 7 comment Find a basis for the set of vectors in $\mathbb{R}^4$ in the subspace (hyperplane) $x_1 +x_2 + 2x_3 + x_4 = 0, x_1 + 2x_2-x_3=0$ It might not be, I could be wrong. I just followed a heuristic I found online, since it doesn't seem to be in my textbook. Nov 7 comment How can I prove a basis (for RowA, NulA, ColA, or NulA^T) with orthogonality conditions? Revised! Sorry about that. I should note that the question I'm reading isn't clear on what orthogonality conditions are, and that is where I'm confused. Thank you for the formatting advice. Nov 7 comment Give an example of a $3$-dimensional subspace of $P_4$ which contains the polynomials… This answer was fantastic, albeit a bit over my head. I believe this is an Einstein Summation? I'm assuming part of what it does can be labeled synonymous with the definition of linear independent if it = 0. I will study until I understand. Nov 7 comment Give an example of a $3$-dimensional subspace of $P_4$ which contains the polynomials… No prob! Just making sure. I'm reading your solution now. Thank you so much for your insight. Nov 7 comment Give an example of a $3$-dimensional subspace of $P_4$ which contains the polynomials… Just curious if I'm missing the obvious, but where does the + 4t^4 come from? Did you combine the first two polynomials?