Is b a linear combination of a1, a2, and a3?

In summary, we are trying to determine if the vector b is a linear combination of the given vectors a1, a2, and a3. This can be done by setting up a linear system with the coefficients A, B, C, and D and solving for them using row reduction. If the only solution is A= B= C= D= 0, then the vectors are independent. However, if there are other solutions, then the vectors are dependent. This process is not affected by the number of variables or whether the matrix is augmented. Additionally, row reduction does not necessarily have to be used to solve equations, it can also be used to determine if a matrix has a unique row reduced form.
  • #1
karush
Gold Member
MHB
3,269
5
$\tiny{311.1.3.12}$
Determine if $b$ is a linear combination of $a_1,a_2$ and $a_3$
$ a_1\left[\begin{array}{r} 1\\0\\1 \end{array}\right],
a_2\left[\begin{array}{r} -2\\3\\-2 \end{array}\right],
a_3\left[\begin{array}{r} -6\\7\\5 \end{array}\right],
b=\left[\begin{array}{r} -7\\13\\4 \end{array}\right]$

ok I don't think this is too difficult to do.
but these matrix problems are very error prone
so thot I would just do a step at a time here
from the example I looked at this is the same thing as
$\left[\begin{array}{lll}a_1&+(-2a_2)&+(-6a_3)\\
&+3a_2 &+7a_3\\
a_1&+(-2a_2)&+5a_3) \end{array}\right]
=\left[\begin{array}{r} -7\\13\\4 \end{array}\right]$
I left all the + signs in since I think this is what a combination is, so then
$\left[\begin{array}{rrr|r}1&-2&-6&-7\\
0&3&7&13\\
1&-2&5&4 \end{array}\right]$
by RREF I got $a_1=3,\quad a_2=2\quad a_3=1$
 
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  • #3
I don't see any matrix!

These vectors will be independent if and only if whenever
\(\displaystyle A\begin{bmatrix}1 \\ 0 \\ 1\end{bmatrix}+ B\begin{bmatrix}-2 \\ 3 \\ -2\end{bmatrix}+ C\begin{bmatrix}-6 \\ 7 \\ 5\end{bmatrix}+ D\begin{bmatrix}-7 \\ 13 \\ 4 \end{bmatrix}= 0\) we must have A= B= C= D= 0.

\(\displaystyle \begin{bmatrix}A- 2B- 6C- 7D \\ 3B+ 7C+ 13D \\ A- 2B+ 5C+ 4D\end{bmatrix}= \begin{bmatrix}0 \\ 0 \\ 0 \end{bmatrix}\).

So we need to solve A- 2B- 6C- 7D= 0., 3B+ 7C+ 13D= 0, and A- 2B+ 5C+ 4D= 0.
If A= B= C= D= 0 is the only solution the vectors are independent. If there are other solutions they are dependent.

Of course, if we were really clever we would have said, right at the start, that four vectors in three dimensional space CAN'T be independent!

No nasty matrices!
 
  • #4
well that's good to know😎
 
  • #5
wanted to add this true or false statement and justification:

Each matrix is row equivalent to one to one and only one reduced echelon matrix
ok I don't think this is true because some matrix are not one to one

The row reduction algorithm applies only to augmented matrices for a linear system
no really sure isn't an augmented matrix Ax=b in one place
 
  • #6
I have no idea what you mean by a matrix being "one to one".

In any case, this question has nothing to do with "one to one". Also row reduction can be applied to any matrix. The matrix doesn't have to be "augmented" and row reduction is not only used to solve equations.

This question is only asking if a row reduction is "unique"- if a given matrix has only one row reduced form..
 
  • #7
I thot one to one meant like 3 variables 3rows 3cols in Ax=b
 
  • #8
No, "one to one" means that each value of the independent variable gives one unique value of the dependent value. It does NOT necessarily have anything to do with matrices or linear transformations.
 

1. What is a linear combination?

A linear combination is a mathematical operation that involves multiplying each term in a set of numbers by a constant and then adding them together. It is commonly used in algebra and linear algebra to solve equations and find solutions.

2. How is linear combination used in science?

Linear combination is used in science to model and analyze complex systems. It is particularly useful in fields such as physics, chemistry, and engineering to describe the relationship between different variables and predict outcomes.

3. What is the significance of 311.1.3.12 in linear combination?

311.1.3.12 is a specific numerical value that represents the coefficients used in a linear combination. These coefficients determine the weights of each term in the combination and can greatly impact the resulting solution.

4. Can linear combination be applied to non-numerical data?

Yes, linear combination can be applied to non-numerical data by converting it into numerical form. This can be done by assigning numerical values to different categories or using techniques such as one-hot encoding.

5. How does linear combination differ from other mathematical operations?

Linear combination differs from other mathematical operations such as addition, subtraction, multiplication, and division in that it involves both multiplication and addition. It also allows for the use of constants, making it a more flexible and powerful tool for solving equations and analyzing data.

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