// Numbas version: finer_feedback_settings {"name": "General linear combinations of standard basis vectors", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "General linear combinations of standard basis vectors", "tags": [], "metadata": {"description": "

Abstract linear combinations. \"Surreptitious\" preview of bases and spanning sets, but not explicitely mentioned. There is no randomisation because it is just an abstract question. For counter-examples, any valid counter-example is accepted.

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This question is about general linear combinations.

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Part a) Whatever real numbers the entries \\(x\\) and \\(y\\) are, we can use those numbers as coefficients. So in general, we have \\(\\begin{pmatrix}x\\\\y\\end{pmatrix}=x\\begin{pmatrix}1\\\\0\\end{pmatrix}+y\\begin{pmatrix}0\\\\1\\end{pmatrix}\\).

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Part b) Similarly, we have \\(\\begin{pmatrix}x_1\\\\x_2\\\\x_3\\\\x_4\\end{pmatrix}=x_1\\var{latex(e1)}+x_2\\var{latex(e2)}+x_3\\var{latex(e3)} + x_4\\var{latex(e4)}\\). Therefore, there is no counterexample.

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Part c) We cannot write any vector in \\(\\mathbb{R}^4\\) as a linear combination of just the first three vectors: any vector which has a non-zero fourth entry cannot be written like this, because

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\\[ x_1\\var{latex(e1)}+x_2\\var{latex(e2)}+x_3\\var{latex(e3)} =\\begin{pmatrix}x_1\\\\x_2\\\\x_3\\\\0\\end{pmatrix}.\\]

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Whatever we choose for \\(x_1,\\ x_2,\\ x_3\\), we can never get a non-zero entry in the fourth position. For example, \\(\\begin{pmatrix}0\\\\0\\\\0\\\\1\\end{pmatrix}\\) is a counter-example. Any vector with non-zero fourth entry is accepted as a correct counter-example.

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first standard basis vector

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Can you write any vector \\(\\begin{pmatrix}x\\\\y\\end{pmatrix}\\in \\mathbb{R}^2\\) as a linear combination of the vectors \\(\\begin{pmatrix}1\\\\0\\end{pmatrix},\\ \\begin{pmatrix}0\\\\1\\end{pmatrix}\\)?

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If yes, fill in the coefficients. If no, write NA into the gaps.

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\\(\\begin{pmatrix}x\\\\y\\end{pmatrix}=\\ \\) [[0]] \\(\\begin{pmatrix}1\\\\0\\end{pmatrix} + \\) [[1]] \\(\\begin{pmatrix}0\\\\1\\end{pmatrix}\\).

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Can you write any vector \\(\\begin{pmatrix}x_1\\\\x_2\\\\x_3\\\\x_4\\end{pmatrix}\\in \\mathbb{R}^4\\) as a linear combination of the vectors \\(\\var{latex(e1)},\\ \\begin{pmatrix}0\\\\1\\\\0\\\\0\\end{pmatrix},\\ \\begin{pmatrix}0\\\\0\\\\1\\\\0\\end{pmatrix},\\ \\begin{pmatrix}0\\\\0\\\\0\\\\1\\end{pmatrix}\\)?

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If yes, fill in the coefficients. If no, write NA into the gaps.

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\\(\\begin{pmatrix}x_1\\\\x_2\\\\x_3\\\\x_4\\end{pmatrix}=\\ \\) [[0]] \\(\\var{latex(e1)}+ \\) [[1]] \\(\\var{latex(e2)}+\\) [[2]] \\(\\var{latex(e3)}+ \\)[[3]]\\(\\var{latex(e4)} \\).

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If you decided that not every vector \\(\\begin{pmatrix}x_1\\\\x_2\\\\x_3\\\\x_4\\end{pmatrix}\\in \\mathbb{R}^4\\) can be written as a linear combination as above, give an example of a vector which cannot be written like this. Write it into the gap as vector(1,2,3,4) with the numbers substituted by your chosen entries. (The marking algorithm will not understand fractions.) If you did provide general coefficients above, then write NA into the gap.

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Counterexample: [[4]]

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Can you write any vector \\(\\begin{pmatrix}x_1\\\\x_2\\\\x_3\\\\x_4\\end{pmatrix}\\in \\mathbb{R}^4\\) as a linear combination of just the first three vectors \\(\\var{latex(e1)},\\ \\begin{pmatrix}0\\\\1\\\\0\\\\0\\end{pmatrix},\\ \\begin{pmatrix}0\\\\0\\\\1\\\\0\\end{pmatrix}\\)?

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If yes, fill in the coefficients. If no, write NA into the gaps.

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\\(\\begin{pmatrix}x_1\\\\x_2\\\\x_3\\\\x_4\\end{pmatrix}=\\ \\) [[0]] \\(\\var{latex(e1)}+ \\) [[1]] \\(\\var{latex(e2)}+\\) [[2]]\\(\\var{latex(e3)}\\).

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If you decided that not every vector \\(\\begin{pmatrix}x_1\\\\x_2\\\\x_3\\\\x_4\\end{pmatrix}\\in \\mathbb{R}^4\\) can be written as a linear combination as above, give an example of a vector which cannot be written like this. Write it into the gap as vector(1,2,3,4) with the numbers substituted by your chosen entries. (The marking algorithm will not understand fractions.) If you did provide general coefficients above, then write NA into the gap.

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Counter-example: [[3]]

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