311.1.7.9 For what values of h is v_3 in Span {v_1,v_2,v_3}

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In summary: The definition of a system of equations is:A system of equations is consistent if every equation in the system can be solved for one of the variables.
  • #1
karush
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$\tiny{311.1.7.9}$
For what values of $h$ is $v_3$ in Span $\{v_1,v_2,v_3\}$ linearly \textit{dependent}. Justify
$v_1=\left[\begin{array}{rrrrrr}1\\-3\\2\end{array}\right],
v_2=\left[\begin{array}{rrrrrr}-3\\9\\-6\end{array}\right],
v_3=\left[\begin{array}{rrrrrr}5\\-7\\h\end{array}\right]$

$v_3$ is in Span{v1, v2} means there exists a constant such that

$c_1v_1 + c_2v_2 = v_3$

So if, but this is an augmented matrix

$\left[\begin{array}{rr|r}1 &−3& 2 \\ −3 &9 &−7 \\5 &−7& h \end{array}\right]$
RREF
$\left[ \begin{array}{ccc} 1 & -3 & 2 \\0 & 0 & -1 \\0 & 8 & h - 10 \end{array} \right]$
anyway, so far:unsure:$
 
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  • #2
Your wording is a little off- it is not a vector in the span that is independent, it is the set of vectors themselves. And to do this problem you need to think about the DEFNITION of "independent vectors": these vectors are independent if and only there do NOT exist numbers, a, b, and c, not all 0, such that \(\displaystyle a\begin{bmatrix}1 \\ -2 \\ 3\end{bmatrix}+ b\begin{bmatrix}-3 \\ 9 \\ -6 \end{bmatrix}+ c \begin{bmatrix}5 \\ -7 \\ h \end{bmatrix}= \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix}\)

That is the same as saying
a- 3b+ 5c= 0
-2a+ 9b- 7c= 0
3a- 6b+ hc= 0

Try to solve those. h will make those vectors independent if the only solution is a= b= c= 0,

That can be done using an augmented matrix similar to your matrix- but your matrix is not "augmented". You need\begin{bmatrix}1 & -3 & 5 & 0 \\ -2 & 9 & -7 & 0 \\ 3 & -6 & h & 0 \end{bmatrix}.
 
  • #3
ok here is the theorm I think we need r3 is a pivit row

Screenshot 2020-12-31 at 4.30.35 PM.png
 
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  • #4
Starting from $\begin{bmatrix}1 & -3 & 5 & 0 \\ -2 & 9 & -7 & 0 \\ 3 & -6 & h & 0 \end{bmatrix}$ add twice the first row to the second row and subtract three times the first row from the third row to get $\begin{bmatrix}1 & -3 & 5 & 0 \\ 0 & 3 & 3 & 0 \\ 0 & 3 & h-15 & 0 \end{bmatrix}$.

Now subtract the new second row from the third row to get $\begin{bmatrix}1 & -3 & 5 & 0 \\ 0 & 3 & 3 & 0 \\ 0 & 0 & h-18 & 0 \end{bmatrix}$.

That matrix is equivalent to the equations a- 3b+ 5c= 0, 3b+ 3c= 0, and (h- 18)c= 0. If h is not 18, we can divide both side of the last equation by h- 18 to get c= 0. Then the second equation becomes 3b= 0 so b= 0 and the first equation becomes a= 0. That is, if h is anything other than 18, the vectors are independent. If h= 18, however, (h- 18)c= 0 becomes 0= 0 which is true for any c, 3b+ 3c= 0 gives b= -c and a- 3b+ 5c= a+ 8c= 0 so a= -8c. There exist infinitely many a, b, c that make the equations true so the vectors are dependent.

I don't know why you are giving the definition of a system of equations being "consistent". The exercise in your first post says nothing about a system of equation nor "consistency".
 

Related to 311.1.7.9 For what values of h is v_3 in Span {v_1,v_2,v_3}

1. What does the notation "v_3 in Span {v_1,v_2,v_3}" mean?

The notation "v_3 in Span {v_1,v_2,v_3}" means that v_3 is a linear combination of v_1, v_2, and v_3. In other words, v_3 can be written as a sum of scalar multiples of v_1, v_2, and v_3.

2. What does the number 311.1.7.9 represent in this context?

The number 311.1.7.9 is likely a specific problem or exercise number in a linear algebra textbook or course. It does not have any particular significance in relation to the question about Span and linear combinations.

3. How is the value of h related to the span of v_1, v_2, and v_3?

The value of h determines whether or not v_3 is in the span of v_1, v_2, and v_3. If h is equal to 0, then v_3 is not in the span. If h is any other value, then v_3 is in the span.

4. Can v_3 be in the span of v_1 and v_2 if h is equal to 0?

No, if h is equal to 0, then v_3 cannot be in the span of v_1 and v_2. This is because the span of v_1 and v_2 only includes linear combinations of v_1 and v_2, and v_3 is not included in this span when h is equal to 0.

5. What is the purpose of determining whether v_3 is in the span of v_1, v_2, and v_3?

The purpose of determining whether v_3 is in the span of v_1, v_2, and v_3 is to understand the relationships and dependencies between vectors. It can also help in solving systems of linear equations and understanding the properties of vector spaces.

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