# Basis and Components (Vectors)

• RyanH42
In summary, the basis vectors are linearly independent and the components of ##a=2u_1-u_3## in terms of ##v_1,v_2,v_3## are ##u_1=-2v_1+v_2-v_3##, ##u_2=3v_1-v_2+2v_3##, and ##u_3=4v_1-2v_2+3v_3##.
RyanH42

## Homework Statement

Let ##u_1,u_2,u_3## be a basis and let ##v_1=-u_1+u_2-u_3## , ## v_2=u_1+2u_2-u_3## , ##v_3=2u_1+u_3## show that ##v_1,v_2,v_3## is a basis and find the components of ##a=2u_1-u_3## in terms of ##v_1,v_2,v_3##

## Homework Equations

For basis vecor we need to clarify these vector are lineerly independet.We can understand linearly independent as make a matrix 3x3 and see the take the det of matrx. If det of matrix is not zero means linearly independent.

## The Attempt at a Solution

I take the determinant of ##v_1,v_2,v_3## and I found -1.It means linearly independent.It means they are basis vectors.Then I tried to show ##u_1=(1,0,0)## then I tried write it in terms of ##v## but I thought that there a lot way to do that or maybe wrong.

The answer is ##u_1=-2v_1+v_2-v_3## and ##u_2=3v_1-v_2+2v_3## and ##u_3=4v_1-2v_2+3v_3## and ##a=-8v_!+4v_2-5v_3##
Thanks

The matrix you are looking the determinant of is:
$\begin{bmatrix} -1 & 1 & -1 \\ 1 & 2 & -1 \\ 2 & 0 & 1 \end{bmatrix}$
Which is -1.

Now you have to go the other way around. I mean you know how $v$'s are given in terms of $u$'s... you should actually go to the inverse relation and see how $u$'s are written in terms of $v$'s (so you can make the substitution in alpha)...

If you are lucky you can see what you have to do right away by a few tries of adding/substracting and multiplying with factors the $v_i$.

Try to find $u_1$ in terms of $v_{1,2,3}$.

(do you know about inverse matrices?)

RyanH42
ChrisVer said:
(do you know about inverse matrices?)
Yeah,I know.

you can use the inverse matrix to get the $u_i$ in terms of $v_j$:

$\begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix} = \begin{bmatrix} -1 & 1 & -1 \\ 1 & 2 & -1 \\ 2 & 0 & 1 \end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix}$

To find ##M## here:

$\begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix}= M \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}$

RyanH42
You can try also your way $u_1 = \begin{pmatrix} 1 \\ 0 \\0 \end{pmatrix}$, $u_2 = \begin{pmatrix} 0 \\ 1 \\0 \end{pmatrix}$, $u_3 = \begin{pmatrix} 0\\ 0 \\1 \end{pmatrix}$.

That means you are taking a particular basis for $u_i$. Then the $v$'s are:

$v_1 = \begin{pmatrix} -1 \\ 1 \\ -1 \end{pmatrix}$, $v_2 = \begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$, $v_3 = \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}$.

Now you would like to see how you can add $v_1,v_2,v_3$ in order to get $u_1$. You can try that by solving the equation for $a,b,c$:

$u_1 = av_1+ bv_2 +cv_3 \Rightarrow \begin{pmatrix} 1 \\ 0 \\0 \end{pmatrix}= a \begin{pmatrix} -1 \\ 1 \\-1 \end{pmatrix}+ b\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix} +c \begin{pmatrix} 2 \\ 0 \\ 1 \end{pmatrix}$.
That's 3 equations with 3 unknown variables (a,b,c) so it's solvable:
\begin{align*} -a +b& +2c = 1 \\ a+2b&=0 \\ -a-b&+c =0 \end{align*}

And similarily work for the rest $u_2 \& u_3$. That will be in total 9 equations with 9 unknown parameters.

That's an alternative way of saying that you are taking the inverse matrix.. you just wrote for $M= \begin{bmatrix} a & b & c \\ d & f & e \\ h & w & r \end{bmatrix}$ and you go to determine its elements one by one.

Last edited:
RyanH42
##1=-a+b+2c##
##0=a+2b##
##0=-a-b+c##
then #### ##a=-2b## then If I put this in third equation equation I found ##b=-c## .If I put these info on first equation I get ##1=2b+b-2b##, ##b=1## then ##a=-2## and ##c=-1## Which İt fits the answer.
I understand the matrix way and this way.Matrix way is look like simpler but Its hard to find inverse matrix of 3x3.I used online calculator and I found exact solution.
Thanks my friend.

## What is a vector?

A vector is a mathematical object that represents both magnitude (size) and direction. It is typically represented graphically as an arrow, with the length representing the magnitude and the direction indicating the direction.

## What is the difference between a vector and a scalar?

While a vector represents both magnitude and direction, a scalar only represents magnitude. In other words, a scalar is a single numerical value, while a vector is a combination of magnitude and direction.

## What are the components of a vector?

The components of a vector are the parts that make up the vector in terms of its direction and magnitude. These components are typically represented as x and y values in a 2-dimensional coordinate system, or x, y, and z values in a 3-dimensional coordinate system.

## How do you add two vectors together?

To add two vectors together, you must add their corresponding components. For example, if you have two 2-dimensional vectors with components (x1, y1) and (x2, y2), their sum would be (x1 + x2, y1 + y2).

## What is the significance of the basis of a vector space?

The basis of a vector space is a set of linearly independent vectors that span the entire space. This means that any vector in the space can be represented as a linear combination of the basis vectors. The basis is important because it allows us to represent complex vectors in a simpler and more organized way.

Replies
4
Views
1K
Replies
34
Views
2K
Replies
11
Views
2K
Replies
1
Views
2K
Replies
1
Views
1K
Replies
3
Views
1K
Replies
1
Views
1K
Replies
4
Views
1K
Replies
2
Views
2K
Replies
3
Views
2K