If u=(5,2) and w=(4,-3) write v=(17,-7) as a linear combination of u and w Is th

Question

If u=(5,2) and w=(4,-3) write v=(17,-7) as a linear combination of u and w
Is the set M2,2 with standard operations a vector space
Is the set of integers with standard operations a vector space
If u=(5,2) and w=(4,-3) write v=(17,-7) as a linear combination of u and w
Is the set M2,2 with standard operations a vector space
Is the set of integers with standard operations a vector space

Is the set M2,2 with standard operations a vector space
Is the set of integers with standard operations a vector space

Explanation / Answer

u = (5, 2) and w = (4, -3) and v = (17, -7)

v as a linear combination of u and w :

v = xu + yw

(17, -7) = x( 5, 2) +y( 4, -3)

17 = 5x +2y ----(1)

-7 = 2x - 3y ------(2)

Multiply equation1 by 3 and equation 2 by 2 and subtract:

we get x= 37/19 ; y = 69/19

So, v = 37u/19 + 69w/19

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Is the set of integers with standard operations a vector space ?

check the properties of vector space:

True.The sum of two integers is again an integer.
False. Recall that the scalars that you use for scalar multiplication are elements of some field.We assume it's Q or R. In either case, 1 is in Z, but (1/2)*1=1/2 isn't in Z.
Holds. 0 is an integer.

So, we Z isn't closed under scalar multiplication

The set Z of integers (standard operations) is not a vector space.

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