Question

Determine if the statement is true or false, and justify your answer. Suppose A is a matrix with n rows and m columns. If n < m, then the columns of A span Rn.

Answer 1

Answer:

The question is incomplete as there are given options ;

Q:

Determine if the statement is true or false , and justify your answer . Suppose A is a matrix with n rows and m columns . If n < m, then the columns of A span R True, since there are more columns than rows. O False, since there are not enough columns to span R". True, since every column of A must be a nonzero column, False, since every column of A may be a zero column, True, since every column of A must be a non zero column, False, since every column of A may be a zero column

Step-by-step explanation:

Considering a matrix A with n-rows and m-columns,

given that n is less than m i.e the rows is less than the columns

then the columns of A span Rn? TRUE OR FALSE?

the matrix is of nxm as n is less than m, hence from linear transformation, T will span : Rm towards Rn

the concept of ranking of a matrix is applied here as ranking entails the  number of linearly independent rows or columns vectors in a matrix, in this case

the order is n x m where n is less than m, as such the rank of the matrix is n

So, Rank of MATRIX A is n

To prove if the rows vectors are linearly independent or not since n is the rank of the matrix

In this case, m columns vector will be considered with respect to n which is the rank and which is also less than m from the conditions, as such there exist a linearly independent relationship between them which R may be spanned since we know that from ranking, reduction to echelon form comes into play by trying to reduce every element of the column A to zero.

• Trying to reduce to echelon form implies all the element of column A may or may not zero.

from the foregoing, the last option is the correct answer ; False, since every column of A may be a zero column