Let B = (a, b, c) be an ordered basis for a vector space V. Among the statements

Question

Let B = (a, b, c) be an ordered basis for a vector space V. Among the statements below select the correct ones. - Warning: some false answers receive negative marks. Select one or more: -1. The dimension of V is 3. O 2. If X is a vector in V, then XB = (2,-3,7) means that cross out x = 2a-3b + 7c cross out 3. If X is a vector in V, then XB = (2,-3, 7) means that x is a vector in R3. cross out O 4. If xis a vector in V, then XB = (2,-3,7) means that x can be turned into a vector in R3 by converting to standard coordinates cross out 5. If XB = (-4,6, 2) and y =-4a + 6b + 2c, then x = y cross out

Explanation / Answer

Given, B = (a,b,c) is an ordered basis for a vector space V.

Now, 1) the dimension of V is 3.

We know that if V is a vector space and {v1,v2,v3,....,vn} is a basis of V, then the dimension of V is defined by dim(V) = n.

Here, {a,b,c} is a basis of V.

Therefore, dim(V) = 3.

Hence, statement is true.

2) If x is a vector in V, then xB = (2,-3,7) means that x = 2a - 3b + 7c.

Here, x = [xB]B = (2,-3,7)B = 2*a + (-3)*b + 7*c = 2a - 3b + 7c.

Hence, statement is true.

3) If x is a vector in V, then xB = (2,-3,7) means that x is a vector in R3.

Here, dim(V) = 3 and dim(R3) = 3. Therefore, dim(V) = dim(R3) ,i.e., the dimensions of V and R3 are same.

Since, x is a vector in V and dim(V) = dim(R3), therefore, x is a vector in R3.

Hence, statement is true.

4) If x is a vector in V, then xB = (2,-3,7) means that x can be turned into a vector in R3 by converting to standard coordinates.

Here,we get from previous statement that x is a vector in R3.

So, there is no need to turn x into a vector in R3 by converting to standard coordinates.

Hence, statement is false.

5) if xB = (-4,6,2) and y = - 4a + 6b + 2c, then x = y.

Here, xB = (-4,6,2). Then, x = - 4a + 6b + 2c.

We also have, y = - 4a + 6b + 2c.

Therefore, x = y.

Hence, statement is true.

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