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12.1 - Linear dependence

karush

Well-known member
Jan 31, 2012
2,657
Are the vectors
$$\left[
\begin{array}{r}
2\\1\\-2
\end{array}\right]
,\quad
\left[\begin{array}{r}
0\\2\\-2
\end{array}\right]
,\quad
\left[\begin{array}{r}
2\\3\\-4
\end{array}\right]
$$
linearly dependent or linearly independent?
$$\left[ \begin{array}{rrr|r} 2 & 0 & 2 & 0 \\ 1 & 2 & 3 & 0 \\ -2 & -2 & -4 & 0 \end{array} \right]
\sim
\left[ \begin{array}{rrr|r} 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \end{array} \right]$$
I assume this is independent due to trivial answers
 
Last edited:

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
Re: 12.1

By the definition of "linearly dependent", these vectors are linearly dependent if and only if there exist three number, a, b, and c, not all 0, such that
[tex]a\begin{bmatrix}2 \\ 1 \\ -2 \end{bmatrix}+ b\begin{bmatrix}0 \\ 2 \\ -2 \end{bmatrix}+ c\begin{bmatrix}2 \\ 3 \\ -4\end{bmatrix}= \begin{bmatrix}2a+ 2c \\ a+ 2b+ 3c \\ -2a- 2b- 4c \end{bmatrix}= \begin{bmatrix}0 \\ 0 \\ 0 \end{bmatrix}[/tex].

That is, 2a+ 2c= 0, a+ 2b+ 3c= 0, -2a- 2b- 4c= 0. From 2a+ 2c= 0, c= -a so the last two equations are a+ 2b- 3a= 2b- 2a= 0 and -2a- 2b+ 4c= 2a- 2b= 0. Both of those give a= b. Any numbers a, b, and c, such that b= a, c= -a will work.

So, yes, these vectors are linearly dependent.

You should think about two questions. What definition of "linearly independent" and "linearly dependent" did you learn? And what was your purpose in row reducing a matrix having the vectors as columns if you had to ask if the vectors were linearly dependent when you finished?
 

karush

Well-known member
Jan 31, 2012
2,657
Re: 12.1

I pretty much just followed an example
But probably could solve some of these just by observation
they used augmented matrix but only did some alteration
View attachment 9045
 
Last edited:

MarkFL

Administrator
Staff member
Feb 24, 2012
13,736
I edited the thread title. The original title of "12.1" wasn't of much use to describe the topic. :)
 

karush

Well-known member
Jan 31, 2012
2,657
Thank you

I tried also to change it

But when I submitted it didn't happen