If A is a 3x5 matrix, explain why the columns of A must
be linearly independent.
Correct! (Yes) (You might perhaps have added that the rank is also the number of linearly independent rows in A, which is why the rank is at most 3.)Yuuki said:i thought if A is 3x5, the columns of A must be linearly dependent, since
the rank is at most 3, and the rank is the number of linearly independent columns in A.
but there are 5 columns in A, so the columns of A must be linearly dependent :/
Column space refers to the set of all possible linear combinations of the columns in a matrix. It represents the span of the columns and is a fundamental concept in linear algebra.
Column space is important because it helps us understand the relationships between the columns in a matrix and can provide insights about the properties of the matrix. It is also used in solving systems of linear equations and in finding solutions to matrix equations.
To calculate the column space of a matrix, we use a process called Gaussian elimination, where we reduce the matrix to its row-echelon form. The columns with leading non-zero entries in the row-echelon form form the basis for the column space.
The dimension of the column space is equal to the number of linearly independent columns in the matrix. This can be determined by counting the number of leading non-zero entries in the row-echelon form of the matrix.
The column space of a matrix is closely related to its null space and row space. The column space and row space are orthogonal complements, and the dimension of the column space and null space sum up to the number of columns in the matrix. Additionally, the column space and null space together form the entire vector space that the matrix operates on.