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Home Q & A Board Science-Related Homework Help Physics
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tony123 tony123
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11 years ago
Say I have some m x n matrix where each column is a vector and am given some random vectors. How do I decide which of those random vectors would be in the column space of this matrix? I understand that the column space of the matrix will be the span of the vectors in that matrix, but other than that I am lost. It would be nice if someone could tell me how you would do it with row space, too.
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#1
11 years ago
Let's say you have a matrix "A" that's m x n.  The "n" column vectors make up a vector space.  Now let's multiply this matrix by some n x 1 vector x = .  The resulting vector ("b") is a linear combination of the column vectors, specifically
b = Ax = x1*col1 + x2*col2 + ... + xn*coln

Now, if we have another vector, "z", and we want to know if it is in the column space of "A", we have to solve the system
z = Ax

for the vector "x"; if there is a solution, then we can write "z" as a linear combination of the columns of "A", and your random vector "z" is in the column space of "A".
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