Question

plz some explain the column space....i am new in linear algebra

Answer 1

It's the vector space spanned by the columns of a given matrix interpreted as vectors (similarly, the row space is spanned by the rows of the matrix, the null space is the vectors which the matrix sends to 0, and the null space of the transpose is self explanatory). Be aware that the row space and nullspace of a given matrix are orthogonal, as are the column space and the null space of the transpose.

Answer 2

Take each column of a matrix and consider it as a vector. If you have a 4 by 5 matrix you have 5 columns, therefore five vectors, each with four entries. Assume v1, v2, v3, v4, and v5 are the five vectors. Then take each of the column vectors, multiply it by some scalar value (which can be different for each vector), and add. For example: c1v1 + c2v2 + c3v3 + c4v4 + c5v5. The result is a vector. If you take the set of all resulting vectors for all possible values of c1 through c5 then that is the column space.

Answer 3

To add to my previous answer and expand on mboorstin's answer: If you take a set of vectors, multiply each by a scalar value, and add, that's known as a linear combination of the vectors. So if v1 through v5 are column vectors then 2v1 + v2 + 3.4v3 - 3v4 + 0v5 is a linear combination of v1 through v5.

To say a set of vectors v1, v2, ..., vn "span" a space means that any vector w in the space can be expressed as a linear combination of v1 through vn. The column space is the set of all linear combinations of the column vectors, so (as mboorstin wrote) the column space of a given matrix is the vector space spanned by the columns of a that matrix interpreted as vectors.

Answer 4

thankx sir