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luda_lava [24]
2 years ago
15

A denotes an mn matrix. Determine whether the statement is true or false. Justify your answer. The row space of AT is the same a

s the column space of A.
Mathematics
1 answer:
ziro4ka [17]2 years ago
4 0

Answer: true

Step-by-step explanation:

For an m*n matrix, the column space of A will be a space formed by the lineal combination of all the columns of A.

column space = a1*c1 + a2*c2 + ...

                       

where a1, a2, ... are scalars, and c1 is the vector of column 1.

Then we should write:

Column space = a1*(A₁₁, A₂₁, A₃₁, ...) + a2*(A₁₂, A₂₂, A₃₂, ...) + ...

Now, the transpose is defined as:

[At]₁₃ = A₃₁

Here i used the element with subindex 3 and 1, but is the same for every subindex.

Notice that if A is m*n, then [At] is n*m

Now, the row space of [At] will be, same as before.

Row space = b1*r1 + b2*r2 + ...

Where b1, b2, ... are scalars and the r's are the vector of each row.

                   = b1*( [At]₁₁ , [At]₁₂, [At]₁₃, ...) + b2*([At]₂₁, [At]₂₂, [At]₂₃, ...) + ...

Now we replace each term of the transpose by the associated element in the original matrix.

                   = b1*( A₁₁, A₂₁, A₃₁, ...) + b2*(A₂₁, A₂₂, ...) + ....

If we take:

b1 = a1, b2 = a2, b3 = a3, ...

We will have that the row space of [At] is the same as the column space of A.

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Answer:

Option C.

Step-by-step explanation:

The given expression is

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We need to find the factored form of the given expression.

The given expression can be rewritten as

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It is clear that \frac{2}{3} is common.

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Hence, the correct option is C.

7 0
3 years ago
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Will mark the BRANLIEST.
True [87]

Answer:

see below

Step-by-step explanation:

Let r = number of rides

total amount of money spent has to be less than  or equal to 50

costs are admission and food and rides

rides cost 1.50 each

50≥ admission + food + rides

50 ≥ 15 +10 + 1.50r

Combine like terms

50 ≥25 + 1.50 r

Subtract 25 from each side

50-25 ≥25-25 + 1.50 r

25  ≥ 1.50 r

Divide by 1.5 on each side

25/1.5 ≥ 1.5r/1.5

50/3  ≥ r

Changing this to a mixed number

16 2/3  ≥r

We can only take a whole number of rides

r = 16

Beverly has not spent all of her 50 dollars since there was a fraction for the rides

cost =  15 +10 + 1.50r

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Step-by-step explanation:

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3 years ago
Suppose xy &gt; 0. Describe the points whose coordinates are solutions to the inequality.
Elan Coil [88]

The solutions fo the inequality are all the points (x, y) that meet these 3 conditions.

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<h3>Which points are solutions of the inequality?</h3>

We want to find points of the form (x, y) that are solutions of the inequality:

x*y > 0

Clearly x and y must be different than zero.

So, for example if x = -1, y can be any negative number, for example y= -3

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This is true.

Now if x  = 1, y must be positive. LEt's take y = 103, then:

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Then we have 3 conditions:

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The solutions fo the inequality are all the points (x, y) that meet these 3 conditions.

If you want to learn more about inequalities:

brainly.com/question/25275758

#SPJ1

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