Why Columns Of A Matrix With M Rows And N Columns Can’T Span R^M For M>N?

HW 3: If a matrix has m rows and n columns, with m > n, the columns of the matrix cannot span R^m.

TRUE. If A has more rows than columns, then there cannot be a pivot in every row.

If a matrix has m rows and n columns, with m > n, the columns of the matrix cannot span R^m. This is because R^m has m dimensions, which means it requires m linearly independent vectors to span R^m. However, since the number of columns in the matrix is n, and n < m (given that m > n), there are not enough vectors in the matrix to span all of R^m.

Furthermore, it is important to note that the rank of a matrix (which is the number of linearly independent rows or columns) cannot exceed the smaller of m and n. If m > n, then the rank of the matrix cannot exceed n, which further limits the ability of the columns to span R^m.

As an example, consider a 3×2 matrix A:

A = [1 2; 3 4; 5 6]

The columns of this matrix are [1 3 5] and [2 4 6]. Since this is a 3×2 matrix (m=3, n=2), there are not enough vectors to span R^3. In fact, the rank of the matrix is only 2 (since there are only 2 linearly independent columns), which is less than the minimum of m and n.

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