How many pivot columns must a 5x7 matrix have if its columns span R^5? Why?
To determine the number of pivot columns a matrix must have if its columns span R^5, we can use the concept of the rank of a matrix.
The rank of a matrix is defined as the maximum number of linearly independent columns or rows in a matrix. In other words, it represents the dimension of the vector space spanned by the columns or rows of the matrix.
According to the Rank-Nullity theorem, for any matrix A, the sum of its rank and nullity is equal to the number of columns in A. In this case, since the matrix spans R^5 (a 5-dimensional space), the rank of the matrix would be 5.
Now, let's consider a 5x7 matrix. The maximum number of pivot columns it can have is 5, which is equal to the rank of the matrix. Having more than 5 pivot columns would mean that the matrix has linearly dependent columns, which would contradict the fact that the columns span R^5.
However, it is also possible for a 5x7 matrix to have fewer than 5 pivot columns. In such cases, the columns of the matrix would not span R^5 completely, but only a lower-dimensional subspace of R^5.
Therefore, to ensure that the columns of a 5x7 matrix span R^5, the matrix must have exactly 5 pivot columns.