Final answer:
A 4x3 matrix can be linearly independent or dependent. A 5x7 matrix must have at least 5 pivot columns for its columns to span in R^5. The statement that the columns of any 4x5 matrix are linearly dependent is false.
Step-by-step explanation:
A 4x3 matrix can be linearly independent or dependent, depending on the entries of the matrix. In general, a matrix is linearly independent if its columns cannot be expressed as a linear combination of the other columns. To determine if a 4x3 matrix is linearly independent or dependent, you need to perform certain tests, such as finding the determinant or row-reducing the matrix.
A 5x7 matrix must have at least 5 pivot columns for its columns to span in R⁵. The pivot columns are the columns that contain a leading 1 after performing row reduction on the matrix. Since R⁵ has 5 dimensions, a 5x7 matrix with fewer than 5 pivot columns would not be able to span in R⁵.
The statement that the columns of any 4x5 matrix are linearly dependent is false. The columns of a 4x5 matrix can be linearly independent, depending on the specific entries of the matrix. The linear independence or dependence of a matrix depends on the entries, not just the size of the matrix.