Expanding on Andy Putnam's Putman's comment, beginners in linear algebra often think of all finite-dimensional vector spaces (over $\mathbb{R}$, say) as the same, in particular as $\mathbb{R}^n$ with the standard basis. This can lead to mistakes in any kind of computation, say in multivariable calculus, where one has to compute simultaneously with a row vector and a column vector (e.g. an element of a vector space together with an element of its dual). The problem is that matrices act differently on row and column vectors, hence if you end up changing basis you need to deal with row and column vectors in opposite ways.
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Expanding on Andy Putnam's comment, beginners in linear algebra often think of all finite-dimensional vector spaces (over $\mathbb{R}$, say) as the same, in particular as $\mathbb{R}^n$ with the standard basis. This can lead to mistakes in any kind of computation, say in multivariable calculus, where one has to compute simultaneously with a row vector and a column vector (e.g. an element of a vector space together with an element of its dual). The problem is that matrices act differently on row and column vectors, hence if you end up changing basis you need to deal with row and column vectors in opposite ways. |
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