Lecture 10 Why do all bases have the same number of vectors If the vectors are dependent they do not form a bases Example 3 x 4 matrix If we want to show the column vectors must be dependent the nullspace is larger than the zero vector There can be at most three pivots meaning there is certainly a free column meaning there is a special solution General statement Every m x n matrix has a nullspace greater than just the zero vector if m n This also proves that the column vectors are dependent When does Ax b not have a solution for every b The column space of the matrix is not all of R The rank of A is not equal to the number of rows The rows of A are dependent The dimension of the column space is not equal to the number of rows The dimension of the row space is not equal to the number of rows
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