Question

Describe the span of the set of columns in the matrix?

Answer 1

\[\left[\begin{matrix}3 & 1 & 4 & 2 \\ 0 & 2 & 3 & 1 \\ 0 & 0 & 0 & 3\end{matrix}\right]\]

Answer 2

How many pivots does this matrix has ?

Answer 3

@ganeshie8 It essentially has 3? and there are 3 rows but it also seems inconsistent

Answer 4

its just a matrix, so the term "inconcistent" does not make sense here

Answer 5

consistentt/inconsistent is a concept related to a system of equations / augmented matrix

Answer 6

heard of the term `rank` before ?

Answer 7

Yeah it means the amount of non-zero terms or related to the number of pivot points

Answer 8

since you have 3 pivots, the rank is 3, right ?

Answer 9

Yep the rank will be 3

Answer 10

doesn't that mean you also have 3 linearly independent column vectors ?

Answer 11

if you remove the 3rd column,

1,2,4 columns are independent right ?

Answer 12

\[\left[\begin{matrix}3 & 1 & 2 \\ 0 & 2 & 1 \\ 0 & 0 & 3\end{matrix}\right]\]

Answer 13

right independent because they do not equal 0

Answer 14

those 3 columns are linearly independent and you can reach every point in 3 dimensional space by `taking linear combinations of those 3 column vvectors`, yes ?

Answer 15

Yeah but why are we ignoring the 3rd column?

Answer 16

beause its an useless column, it can be obtained by taking linear combinations of first columns

Answer 17

Oh alright nice that makes more sense so by only using the other 3 we can definitely take linear combinations of the column vectors

Answer 18

Yes,

\[\left[\begin{matrix}3 & 1 & 2 \\ 0 & 2 & 1 \\ 0 & 0 & 3\end{matrix}\right] \pmatrix{x\\y\\z} = \pmatrix{b_1\\b_2\\b_3}\]

has a solution for all vectors of right hand side

Answer 19

So the given four vectors span a full 3 dimensional space

Answer 20

`span` is just a short hand form for saying a long sentence :
`what all vectors can be obtained by taking linear combinations of given set of vectors?`

Answer 21

Yeah so to describe that i can say that by using any linearly combination for columns 1 2 and 4 i can obtain the vectors B1, b2, and b3?

Answer 22

nope

Answer 23

b1, b2, b3 are the components of a vector

Answer 24

And observe that third column is NOT the bad column,
you could have removed 2nd also

Answer 25

1,3,4 column vectors are also linearly independent

Answer 26

but thats not the point here

Answer 27

Not sure what the problem is? would it be the last column? if we remove that then we have a row of zeros

Answer 28

you may describe `span` of given matrix like this :
The given matrix has 3 pivots, that makes the rank = 3, and so it has 3 linearly independent column vectors.

3 linearly independent column vectors span a 3 dimensional space. So the given FOUR column vectors span 3 dimensional space

Answer 29

You don't need to delete any column vector. I have deleted just for explaining you. thats all :)

Answer 30

Oh alright

Answer 31

the definition you gave is awesome it makes sense but the 4 vectors to 3 dimensional space doesn't right now

Answer 32

how many vectors do you want in a 3 dimensional space ?

Answer 33

3

Answer 34

and how do you know the given vectors belong to a 3 dimensional space ?

Answer 35

you need a MINIMUM of 3 linearly independent vectors to span 3 dimensional space, but you can have ANY number of columns in your matrix right ? why restrict it to 3 ?

Answer 36

Not sure

Answer 37

In general :
"n" linearly indpendent vectors span "n" dimensional space

Answer 38

So if you have "4" vectors in "3" dimensional space, they will be DEPENDENT for sure

Answer 39

Okay so if there are more vectors than the dimensional space this makes it dependent?

Answer 40

wouldn't i have a trivial solution though since the last column only has 1 solution?

Answer 41

if that doesnt make sense, consider 3 vectors in xy plane :