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bearhunter [10]
3 years ago
13

Every column of AB is a combination of the columns of A. Then the dimensions of the column spaces give rank(AB) ≤ rank(A). Probl

em: Prove also that rank(AB) ≤ rank(B)
Mathematics
1 answer:
Juliette [100K]3 years ago
3 0

Answer:

The proof is given below:

Step-by-step explanation:

We recall that the rank of a matrix MM is the dimension of the range R(M)R(M) of the matrix MM.

So we have

rank(AB)=dim(R(AB)),rank(A)=dim(R(A)).rank(AB)=dim(R(AB)),rank(A)=dim(R(A)).

In general, if a vector space VV is a subset of a vector space WW, then we have

dim(V)≤dim(W).dim (V)≤dim(W).

Thus, it suffices to show that the vector space R(AB)R(AB) is a subset of the vector space R(A)R(A).

Consider any vector y∈R(AB)y∈R(AB). Then there exists a vector x∈Rlx∈Rl such that y=(AB)xy=(AB)x by the definition of the range.

Let z=Bx∈Rnz=Bx∈Rn.

Then we have

y=A(Bx)=Azy=A(Bx)=Az

and thus the vector yy is in R(A)R(A). Thus R(AB)R(AB) is a subset of R(A)R(A) and we have

rank(AB)=dim(R(AB))≤dim(R(A))=rank(A) [Proved]

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zhuklara [117]

Answer:

Check the solution in both equations. The solution is ( − 1, 2). Solve the system by graphing: {− x + y = 12 x + y = 10 . Solve the system by graphing: {2x + y = 6 x + y = 1 . In all the systems of linear equations so far, the lines intersected and the solution was one point.

Hope it helps!!!

6 0
2 years ago
(9•4)•18 = 9•(4•18)<br><br><br> A True <br> B. False
AveGali [126]

Answer:

false

Step-by-step explanation:

7 0
3 years ago
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Which description best compares the graphs given by the equations:
nexus9112 [7]

x - 5y = -5, -5x - 25y = 25

First, you'll need to get the x variable by itself.

x - 5y = -5<u>
</u><u>    +5    +5</u><u>
</u>       x = 0
So x is plotted on the 0.

For the second part of the first equation, you'll be looking for what the y variable represents.

x - 5y = -5
<u>-x           -x</u><u>
</u>     <u>-5y</u> = <u>-5</u><u>
</u><u>       5      5</u><u>
</u>        y = 1
So y is plotted on the 1 on the vertical line above the 0.

For the first part of the second equation, you'll do the same thing as in the first equation.

-5x - 25y = 25
<u>        +25   +25</u><u>
</u>          <u>-5x</u> = <u>50</u><u>
</u>            5      5
            x = 10
So the x for this equation is plotted on 10 on the horizontal line.

For the second part of the second equation, you will do the same thing as in the first equation.

-5x - 25y = 25
<u>+5               +5</u><u>
</u>       <u>-25y</u> = <u>30</u><u>
</u>        25      25
           y = 1.2
So the y for the second half of the second question is plotted on 1.2 on the vertical line.

<h2>Answer: B) Perpendicular</h2>

Perpendicular means the lines may or may not be of equal length and they will not be perfectly in line with each other.

Parallel means the lines may or may not be of equal length but will be perfectly in line with each other.

Intersecting means the lines may or may not be of equal length but will touch each other.

6 0
2 years ago
Solve this problem on paper using all four steps. A girl scout troop sold cookies. If the girls sold 5 more boxes the second wee
Varvara68 [4.7K]
This is a problem where you should make an equation.
Let's say x is the first week. It would mean
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Third Week : 2(x+5)
You need to add these up to make your equation. Your equation will look something like this :
x+x+5+2(x+5) = 431
Now you simplify.
2x+5+2x+10 = 431
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4x = 416
x = 104
That means they sold 104 Boxes in the First Week.
104+5=109
So they sold 109 Boxes the Second Week.
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And they sold 218 Boxes the Third Week.

I hope I was useful!
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sleet_krkn [62]
A.

First you have to evaluate (2000* .045) then you take that and multiply by 8 then you add the answer to 2000.

so 2720 is your answer.
5 0
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