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katrin2010 [14]
3 years ago
11

Given vectors u = (−1, 2, 3) and v = (3, 4, 2) in R 3 , consider the linear span: Span{u, v} := {αu + βv: α, β ∈ R}. Are the vec

tors (2, 6, 6) and (−9, −2, 5) in Span{u, v} ?
Mathematics
1 answer:
julia-pushkina [17]3 years ago
4 0

Answer:

(2,6,6) \not \in \text{Span}(u,v)

(-9,-2,5)\in \text{Span}(u,v)

Step-by-step explanation:

Let b=(b_1,b_2,b_3) \in \mathbb{R}^3. We have that b\in \text{Span}\{u,v\} if and only if we can find scalars \alpha,\beta \in \mathbb{R} such that \alpha u + \beta v = b. This can be translated to the following equations:

1. -\alpha + 3 \beta = b_1

2.2\alpha+4 \beta = b_2

3. 3 \alpha +2 \beta = b_3

Which is a system of 3 equations a 2 variables. We can take two of this equations, find the solutions for \alpha,\beta and check if the third equationd is fulfilled.

Case (2,6,6)

Using equations 1 and 2 we get

-\alpha + 3 \beta = 2

2\alpha+4 \beta = 6

whose unique solutions are \alpha =1 = \beta, but note that for this values, the third equation doesn't hold (3+2 = 5 \neq 6). So this vector is not in the generated space of u and v.

Case (-9,-2,5)

Using equations 1 and 2 we get

-\alpha + 3 \beta = -9

2\alpha+4 \beta = -2

whose unique solutions are \alpha=3, \beta=-2. Note that in this case, the third equation holds, since 3(3)+2(-2)=5. So this vector is in the generated space of u and v.

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9514 1404 393

Answer:

  (c)  -3/4

Step-by-step explanation:

Subtracting a positive number moves you to the left on the number line. Subtracting a negative number moves you in the opposite direction, to the right.

Here, we start at -2 1/2 = -5/2, and we move 1 3/4 = 7/4 to the right from there. Each mark on this number line is 1/4 unit, so we move 7 marks. The results is ...

  -2 1/2 -(-1 3/4) = -5/2 +7/4

  = -10/4 +7/4 = -3/4

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An isosceles trapezoid with bottom base 26.5 decimeters, top base 8.5 decimeters, and height 9 decimeters. Find the area. a. 238
Tamiku [17]

Answer:

I don't understand can you explain it differently?

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A pitcher contains a liquid mixture of water and lemon juice. The water makes up 2/5 of the weight of the liquid mixture. There
notka56 [123]

Answer:

37.5

Step-by-step explanation:

25÷ 2 is 12.5

so 1/5 of the pitcher is 12.5 ounces

theres 3/5 left to fill in the pitcher

so the answer is 12.5 x 3 which is

37.5

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jan has 500 pieces of paper she prints as many copies as possible of a 16 page report how many pieces of paper are left
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A scale drawing of a rectangular mural has the dimensions 2 inches by 3 inches. The scale is 0.5 inches : 5 feet. Find the actua
Tresset [83]

Answer:

Part A) The actual dimensions of the mural are 20 ft by 30 ft

Part B) The dimensions on the new scale drawing are 0.50 in by 0.75 in

Step-by-step explanation:

Part A) we know that

A rectangular mural has the dimensions 2 inches by 3 inches

The scale of the drawing is \frac{0.5}{5}\frac{in}{ft}

using proportions

<em>Find out the dimensions of the actual mural</em>

For 2 inches

\frac{0.5}{5}\frac{in}{ft}=\frac{2}{x}\frac{in}{ft}\\\\x=5*2/0.5\\x=20\ ft

For 3 inches

\frac{0.5}{5}\frac{in}{ft}=\frac{3}{x}\frac{in}{ft}\\\\x=5*3/0.5\\x=30\ ft

therefore

The actual dimensions of the mural are 20 ft by 30 ft

Part B) Find the dimensions of another scale drawing with the scale 0.25 inches : 10 feet.

To find out the dimensions in the new scale drawing, multiply the actual dimensions by the new scale

The new scale is \frac{0.25}{10}\frac{in}{ft}

For 20 ft

\frac{0.25}{10}\frac{in}{ft}(20\ ft)=0.50\ in

For 30 ft

\frac{0.25}{10}\frac{in}{ft}(30\ ft)=0.75\ in

therefore

The dimensions on the new scale drawing are 0.50 in by 0.75 in

3 0
3 years ago
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