What is the dimension of the vector space consisting of five-by-one column matrices where the rows sum to zero and the first row is equal to the second row? a. 5
b. 4
c. 3
d. 2

Option c.

Step-by-step explanation:

If we have a vector of N components (or variables), and we have K linear independent restrictions for these N components (such that K < N, we can't have more restrictions than components.)

The dimension of the vector will be given by N - K.

Here we know that we have a vector of 5 components, that can be written as:

And we have two restrictions, so we can expect that the dimension of the vector is:

5 - 2 = 3

But let's see it, the restrictions are:

"the first row is equal to the second row"

Then we can rewrite our vector as:

Notice that now we have only 4 variables, v₁, v₃, v₄, and v₅

We also know that the sum of the rows is equal to zero, thus:

v₁ + v₂ + v₃ + v₄ + v₅ = 0

we know that v₂ = v₁, so we can replace that to get:

2*v₁ + v₃ + v₄ + v₅ = 0

Now we can isolate one of the variables, to write it in term of the others, for example, let's isolate v₅:

v₅ = -2*v₁ - v₃ - v₄

Now if we replace that in our vector, we have:

Notice that our vector depends on only 3 variables,  v₁, v₃, and v₄, so we can define our vector in a 3-dimensional space.

Then the correct option is c, the dimension of the vector space is 3.

the first one, all those points on the line just make a strait line.

c. 27 units.

step-by-step explanation:

we are told that in circle a, ∠bae ≅ ∠dae.

we can see from our given diagram that in and ;

, as these are radii of our given circle.

therefore, by sas congruence postulate.

hence, side be will be equal to side de as corresponding parts of congruent triangles are congruent. so we can set an equation to find the value of x as:

now let us substitute x=17 in the expression for the length of be.

therefore, length of be will be 27 units and option c is the correct choice.