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fackert
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Why is it not possible for the columns of a matrix to span R^2 even if those columns do not span R^3?
Is that really what you meant to ask? There is no "why" for a statement that isn't true to begin with! The columns of the matrixfackert said:Why is it not possible for the columns of a matrix to span R^2 even if those columns do not span R^3?
This is because for a set of vectors to span a vector space, they must be linearly independent. If the columns of a matrix span R^2, it means that any vector in R^2 can be written as a linear combination of those columns. However, if the matrix has two columns, it can only span a maximum of two dimensions, which is not enough to cover the entire R^2 space. Therefore, the columns of the matrix cannot span R^2.
No, the columns of a matrix cannot span R^2 even if the matrix has more than two columns. This is because the number of columns in a matrix determines the maximum number of dimensions it can span. So, if the matrix has n columns, it can span a maximum of n dimensions, which is still not enough to span the entire R^2 space.
Yes, in addition to having linearly independent columns, the columns of a matrix must also have the same number of elements as the dimension of the vector space it is trying to span. In the case of R^2, the columns must have two elements. If this condition is not met, the columns of the matrix cannot span R^2.
Yes, it is possible for the columns of a matrix to span a vector space of higher dimensions than the number of columns. This can happen if the columns are linearly dependent, meaning that one or more columns can be written as a linear combination of the others. In this case, the columns may not span the entire vector space, but they can still span a subset of it.
Yes, the order of the columns in a matrix can affect its ability to span a vector space. If the columns are not arranged in a way that allows for linear independence, the matrix may not be able to span the vector space. For example, if the first column is a multiple of the second column, the matrix cannot span the entire vector space, even if it has enough columns to do so.