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## Linear algebra

### Course: Linear algebra>Unit 1

Lesson 3: Linear dependence and independence

# Introduction to linear independence

Introduction to linear dependence and independence. Created by Sal Khan.

## Want to join the conversation?

• Khan says at that three co-planar vectors would be "linearly dependent." Wouldn't they be "planarly dependent", not "linearly dependent?" Doesn't "linear," by definition, mean "line", and "planar", plane? •   The term to use is always "linearly" independent or dependent regardless how many dimensions are involved. I'm not a mathematician, but I am in the class Linear Algebra at college, and we use the same thing. I can say that the terms come from the concept of linear combination which is the addition of vectors in a vector space which are scaled (by multiplication). Sal defines a linear combination in the previous video and says that the reason for the word "linear" is that the focus is on this scaling that takes place - as in, the use of the scalar. I won't try to say I completely understand. To me it is just semantics. After doing enough of this, you're not really thinking of the word linear when you say linearly independent anyway. You're focused on whether or not the linear combination spans the vector space.

It's an interesting question though.
• Wait, so shouldn't the example with 3 vectors in R2 be linearly independent? because they are providing you with the directions you need to span R2 ? • good question. at he partially addresses it. The third vector is unneeded as a basis for R2. Any set of two of those vectors, by the way, ARE linearly independent. Putting a third vector in to a set that already spanned R2, causes that set to be linearly dependent.
• Since you can span all of R^2 with only 2 vectors, does that mean that any set of 3 or more two-dimensional vectors will be linearly dependent? • Yes, since you can span all of R^2 with only 2 vectors, any set of 3 or more vectors in R^2 will be linearly independent! This is because you'll learn later that given any subspace, any basis of that subspace will have the same number of vectors (this number of vectors is called the dimensionality of the subspace) so any set of vectors from that subspace with more vectors than the dimensionality of the subspace will intuitively be linearly dependent (that might not make sense now, but by the time you finish this playlist, it probably will). Good job on figuring that out!
• This may seem a no brainer, but what -is- a dimension, in the mathematical sense? It's one of those concepts that I understand (I think) in my head but cannot explicitly put into words. I can give examples of things in various dimensions, but I cannot yet explain what a dimension really is. • If B is a basis for a vector space V, then the dimension of V is the number of vectors in the basis B.
If you don't know what bases are yet, then an intuitive way to identify dimension of Vector spaces, is to count the number of entries in the vector. For example, R^4 is 4th dimensional because it has 4 entries, the vector space of all 5x6 matrices is 30th dimensional because there are 30 entries in a 5x6 matrix.
• says that span (v1,v2)=R^2, but V3 is linearly dependent, so i am assuming that the set{v1,v2,v3} is linearly dependent but the set{v1,v2} are linearly independent?...just clarifying??..so does this mean only 2 linearly independent vectors can span a vector space? • Is it correct to say that for vectors to be linearly independent they must lie in different dimensions? Is it the only necessary condition? • In case of 3 dimensions, how do I express (calculate) a span of a vector on a surface. Like, if the span is a surface, how do I find and express it? • To express a plane, you would use a basis (minimum number of vectors in a set required to fill the subspace) of two vectors. The two vectors would be linearly independent. So the span of the plane would be span(V1,V2). To express where it is in 3 dimensions, you would need a minimum, basis, of 3 independently linear vectors, span(V1,V2,V3).
• Are the subjects covered by the videos on linear combinations, spans and linear dependence and independence pure math theory? Do they have application to the physical sciences, engineering or computer science? • What happens when linear independence is different between rows and columns?

If I have three "column" vectors in R^4,
1 1 0
0 0 0
1 0 1
0 1 1

I thought because there is a zero row vector, this collection would be linearly dependent.

Why is it linearly independent...?

Thank you! • It is not true that if there is a zero row vector, then the collection is linearly dependent.
Because here is a counter example:
Consider
1 0
0 0
0 1
These column vectors are linearly independent, because one is the i unit vector, and the other one is k, and we know these vectors to be linearly independent (also notice the zero row).
A zero row indicating linearly dependence is only true for matrices that are square, or matrices that are m by n, where m<n (i.e. more columns than rows).
Really the simplest way to check if a set of vectors are linearly independent, is to put the vectors into a matrix, row reduce the matrix to echelon form, then the vectors are linearly independent if and only if there is a pivot in every column. 