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# Column space of a matrix

Introduction to the column space of a matrix. Created by Sal Khan.

## Want to join the conversation?

• why is it called column space when it just the linear combination of vectors,or am i missing something?
• Column space is not as clumsy as "linear combination of vectors". Eventually you'll talk in terms of column space, null space, etc. The word "space" as in "vector space" has it's own meaning and properties so having the word space in the help to remind you it has those properties. I don't think you're not missing anything (unless I am too :))
• HI! Why do you always define the matrix as a bunch of column vectors? Could you also say that elements of each row are components of a row vector or transposed column vector?
Thank you!
• Yes, you could also work out all of linear algebra using row vectors instead of column vectors (but you would have to adapt most of the operations for this). Historically mathematicians have always used column vectors, so we continue to use them that way.

There is no other reason than historical custom to continue doing so. And since everyone in the world will do the same, it's good practice to adapt to the standard.
• Is a Column Space the exactly same thing as the Span?
• No. It's a span, of the column vectors. You can take the span of any set of vectors.
• Why have two names for the same property of a matrix? It seems to me that the span of the column vectors of the matrix is just a second name for the column space of the matrix in all cases. Are there subspaces for which this is not true?
• There are many things that have multiple names that represent the exact same thing, ("antiderivative" and "indefinite integral" comes to mind) it might be because they were discovered/invented by different people or following different methodologies, and it's simpler to just retain all the names than try to unify everything.

You are right, though: "matrix column space" and "span of the column vectors of a matrix", will always represent the exact same thing.
• I'm trying to tell the difference between a column space and a span. This is what I understand so far, can someone tell me if it's correct:
The column space is all the possible vectors you can create by taking linear combinations of the given matrix.
In the same way that a linear equation is not the same as a line, a column space is similar to the span, but not the same. The column space is the matrix version of a span. The Span is the graphical representation of the column space.
• You take the span of a set of vectors. You take the column space of a matrix. The column space of a matrix is the span of its column vectors. Taking the span of a set of vectors returns a subspace of the same vector space containing those vectors.
• I have watched four lessons about Null space and don not understand why to much attention to the simple idea - everything what you multiple on zero make it zero. Really. I do not understand what sens.
• Setting a matrix equal to 0 is a really common thing to do, since 0 has the property of canceling out multiplication and leaving addition the same. There are also other aspects that use the nullspace, you will see more of those as you go through the videos.
• This question does not refer to a time in this video, it actually relates to the topic. If I am of the thinking that the column space is just the span of the matrix, would I be correct?
• Partially. The column space is equal to the span of the set of column vectors that compose a matrix.
• I don't see how you can multiply x1 times v1 if v1 is a column vector with m elements and x1 is a column vector with n elements. How is that possible?
• You aren't. x1 is not a column vector. x1 is the first element of the column vector x. In other words, x1 is a scalar.
• sorry, but my question is more of a general question. I understand all the previous concepts in your videos very well but I feel I need to solve some exercises on my own, can you recommend any exercises or books for me, please ?
• Linear Algebra and Its Applications by David C. Lay has some pretty good exercises.
(1 vote)
• So is Metrix A(V1,V2,...Vn) a valid subpace?