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Magazine Problem 1
Find three bases for \(\mathbb{R}^{2}\) over \(\mathbb{R}\), no two of which have a vector in common.
Problem 2
Determine whether the given set of vectors is a basis for \(\mathbb{R}^{3}\) over \(\mathbb{R}\). $$ \\{(1,1,0),(1,0,1),(0,1,1)\\} $$
Problem 3
Determine whether the given set of vectors is a basis for \(\mathbb{R}^{3}\) over \(\mathbb{R}\). $$ \\{(-1,1,2),(2,-3,1),(10,-14,0)\\} $$
Problem 4
Give a basis for the indicated vector space over the field. $$ Q(\sqrt{2}) \text { over } Q $$
Problem 5
Give a basis for the indicated vector space over the field. $$ \mathrm{R}(\sqrt{2}) \text { over } \mathbb{R} $$
Problem 6
Give a basis for the indicated vector space over the field. $$ Q(\sqrt[3]{2}) \text { over } Q $$
Problem 9
Give a basis for the indicated vector space over the field. $$ Q(\sqrt{2}) \text { over } Q $$
Problem 11
Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication. The vectors in a subset \(S\) of a vector space \(V\) over a field \(F\) span \(V\) if and only if each \(\beta \in V\) can be expressed uniquely as a linear combination of the vectors in \(S\).
Problem 12
Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication. The vectors in a subset \(S\) of a vector space \(V\) over a field \(F\) are linearly independent over \(F\) if and only if the zero vector cannot be expressed as a linear combination of vectors in \(S\).
Problem 13
Correct the definition of the italicized term without reference to the text, if correction is needed, so that it is in a form acceptable for publication. The dimension over \(F\) of a finite-dimensional vector space \(V\) over a field \(F\) is the minimum number of vectors required to span \(V\).
