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bezimeni [28]
2 years ago
14

* Let S = Span {(2,-1, 1), (3, 1, 1), (1, 2, 0)}. (i) Calculate the dimension of S.

Mathematics
1 answer:
Sholpan [36]2 years ago
3 0

The span of 3 vectors can have dimension at most 3, so 9 is certainly not correct.

Check whether the 3 vectors are linearly independent. If they are not, then there is some choice of scalars c_1,c_2,c_3 (not all zero) such that

c_1 (2,-1,1) + c_2 (3,1,1) + c_3 (1,2,0) = (0,0,0)

which leads to the system of linear equations,

\begin{cases} 2c_1 + 3c_2 + c_3 = 0 \\ -c_1 + c_2 + 2c_3 = 0 \\ c_1 + c_2 = 0 \end{cases}

From the third equation, we have c_1=-c_2, and substituting this into the second equation gives

-c_1 + c_2 + 2c_3 = 2c_2 + 2c_3 = 0 \implies c_2 + c_3 = 0 \implies c_2 = -c_3

and in turn, c_1=c_3. Substituting these into the first equation gives

2c_1 + 3c_2 + c_3 = 2c_3 - 3c_3 + c_3 = 0 \implies 0=0

which tells us that any value of c_3 will work. If c_3 = t, then c_1=t and c_2 = -t. Therefore the 3 vectors are not linearly independent, so their span cannot have dimension 3.

Repeating the calculations above while taking only 2 of the given vectors at a time, we see that they are pairwise linearly independent, so the span of each pair has dimension 2. This means the span of all 3 vectors taken at once must be 2.

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Answer:

7.5 quarts of olive oil remain.

Explanation:

Let

q = quarts of olive oil remain

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Multiply both sides by .25 to isolate q.

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3 0
3 years ago
If the volume of a cube is 729cm^3, what is the surfface area.<br><br> Thanks!!
bezimeni [28]

Answer:

486cm^2

Step-by-step explanation:

cube volume: 729cm^3

cube faces count: 6

How you calculate square area:

s(cm) * s(cm) or just s^2(cm^2)

s = side length

If you want to find out the side length of a cube by its volume. Then simply just remove the ^3 by taking the cubic root (also written ^(1/3))

729^(1/3) = 9

then just calculate the area like i showed earlier:

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But that is only one side of a cube's 6 sides, so multiply by 6:

81cm^2 * 6 = 486cm^2

Hope i could help :D

Dont be scared to ask me questions..

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