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## Linear algebra

### Course: Linear algebra>Unit 1

Lesson 3: Linear dependence and independence

# Span and linear independence example

Determining whether 3 vectors are linearly independent and/or span R3. Created by Sal Khan.

## Want to join the conversation?

• if the set is a three by three matrix, but the third column is linearly dependent on one of the other columns, what is the span? A plane in R^3? •  The Span can be either:

case 1: If all three coloumns are multiples of each other, then the span would be a line in R^3, since basically all the coloumns point in the same direction.

case 2: If one of the three coloumns was dependent on the other two, then the span would be a plane in R^3.
• Sal uses the world orthogonal, could someone define it for me? •  Orthogonal is a generalisation of the geometric concept of perpendicular. When dealing with vectors it means that the vectors are all at 90 degrees from each other.
• but two vectors of dimension 3 can span a plane in R^3 • instead of setting the sum of the vectors equal to [a,b,c] (at around )could you not just set the sum of the vectors equal to zero, prove the set's linearly independent and say that implies the span is R3 as none of the vectors are redundant? finding the equations for c1, c2 and c3 seems pointless, unless i missed something • Sean,
I think Sal is trying in this video to relate the concepts of linear independence and span. The earlier videos have covered linear independence and linear dependence.... and they've also covered span. But none of the earlier videos have proved (to my knowledge, anyways) that to span R^n requires at minimum n linearly independent vectors. Also, none of the videos have covered the concept that n linearly independent vectors always spans R^n.
• First. Say i have 3 3-tuple vectors. but they Don't span R3. that is: exactly 2 of them are co-linear. How would I know that they don't span R3 using the equations for a,b and c? (in other words, how to prove they dont span R3 )

Second. How can the equations for a, b and c tell me that the span for the said vectors is in fact R2 ? • In order to show a set is linearly independent, you start with the equation `c₁x⃑₁ + c₂x⃑₂ + ... + cₙx⃑ₙ = 0⃑` (where the x vectors are all the vectors in your set) and show that the only solution is that `c₁ = c₂ = ... = cₙ = 0`. If you can show this, the set is linearly independent. In this video, Sal does this by re-writing the equation as a system of equations. This isn't the only way to do it, but it's the easiest to understand for now.

Let's do an example that does what you want. Given the set:
`x⃑₁ = [1 1 1]`
`x⃑₂ = [1 2 3]`
`x⃑₃ = [2 3 4]`
We want to show if they're linearly independent. So, let's plug it into our original equation (I'm going to use a, b, and c instead of c₁, c₂, and c₃):
`a[1 1 1] + b[1 2 3] + c[2 3 4] = [0 0 0]`
This means that:
`a + b + 2c = 0` (notice the coefficients in columns are the original vectors)
`a + 2b + 3c = 0`
`a + 3b + 4c = 0`
Now we combine our system of equations to see if we can solve for a, b, and c.
` b + c = 0` (found by subtracting line 1 from line 2)
`2b + 2c = 0` (found by subtracting line 1 from line 3)
If we were to continue, we'd try to eliminate the b variable by subtracting the top equation from the bottom twice, but doing so would give us `0 = 0`, so we can't do anything more to simplify. This means that the set is linearly dependent since we can't solve for a, b, or c. Since eliminating just 1 more variable would have solved the system, we know that there's 1 redundant vector in the set and there's therefore 2 linearly independent vectors in the set. The span of 2 LI vectors is always a 2-dimensional subspace of Rn (this is different from spanning R2).
• Linear Algebra starting in this section is one of the few topics that has no practice problems or ways of verifying understanding - are any going to be added in the future? • Does Gauss- Jordan elimination randomly choose scalars and matrices to simplify the matrix isomorphisms • With Gauss-Jordan elimination there are 3 kinds of allowed operations possible on a row.
1) A row can be multiplied by n (n is an arbitrary scalar)
2) A row can be swapped with another row
3) A row can be added to another row or subtracted from another row

You can do multiple steps at once. For example in this video Sal replaces the third row with the third row times 3 - the second row.
If you want to reduce the chance of mistakes you can write this all down in the form of an augmented matrix. (See earlier videos)
• Can anyone give me an example of 3 vectors in R3, where we have 2 vectors that create a plane, and a third vector that is coplaner with those 2 vectors. I can create a set of vectors that are linearlly dependent where the one vector is just a scaler multiple of the other vector.
eg: (-3, -1, 2);(1,2,3);(2,4,6)
But im looking for an example of a set of 3 vectors where the third vector is coplaner with the other 2 vectors, but not just on the same line as one of the vectors. • Two vectors forming a plane: (1, 0, 0), (0, 1, 0).

A third vector coplanar with those but not a multiple of either: (1, 1, 0).

As you see, it's easier to think of this in two dimensions. My first two vectors span the x-y plane, and my third vector is the line y=x. The third dimension doesn't really add anything to the problem.
• is it possible to have 3 linearly independent 2-tuples? • No.

If the three 2-tuples were linearly independent, it would mean that the a 2-tuple could not be expressed as a linear combination of the other two. But since the two are linearly independent, the third 2-tuple can be expressed with the other two, which is a contradiction.

tl;dr: You need two and only two 2-tuples to span R^2, any more would make the set linearly dependent. 