How do I work out the dimension of a subspace?

Question

Isn't the dimension just how many vectors there are? But I'm confused, because given W=span{(0, 2, 3) , (0, 1, -2), (1, 5,-7)}, I'd say the dimension is three, but it can't be that easy!

Same for V=span{(1, 0, -1), (2, 1, 1) , (0, 1, 3)}? How do I find the dimension for each of these?

Thank you so much for any help!

Answer 1

The dimension of a linear space V is the maximum number of linearly independent vectors you can have in a subset of V(ie. the number of vectors in a given basis). Now, in the case of

W = span{(0, 2, 3) , (0, 1, -2), (1, 5,-7)},

since any element of W is a linear combination of the vectors in

{(0, 2, 3) , (0, 1, -2), (1, 5,-7)},

we know that the dimension is *at most* three. In order to show that the dimension is three, we need to prove that these vectors are linearly independent (ie. they form a basis for W). This is easily done: if the scalars a, b and c satisfy

a.(0, 2, 3) + b.(0, 1, -2) + c.(1, 5,-7),

then, by comparing coefficients, we have that

c = 0,
2a + b + 5c = 0,
3a - 2b - 7c = 0;

solving this system of equations yields a = b = c = 0, and so the vectors are linearly independent, and therefore they form a basis for W. Hence, dim(W) = 3.

***

For

V = span{(1, 0, -1), (2, 1, 1) , (0, 1, 3)},

note that

2.(1, 0, -1) - (2, 1, 1) + (0, 1, 3) = (0, 0, 0),

so these vector are not linearly independent. Thus, we know that a basis for V must have *two* vectors or less. So, getting rid of (0, 1, 3), consider the set {(1, 0, -1), (2, 1, 1)}. This is a linearly independent set because, if scalars a and b satisfy

a.(1, 0, -1) + b.(2, 1, 1) = (0, 0, 0),

we have that

a + 2b = 0,
b = 0,
b - a = 0,

and solving this system yields that a = b = 0. Thus V has a basis of 2 vectors, and so dim(V) = 2.

Answer 2

Dimension Of Subspace

Answer 3

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RE:
How do I work out the dimension of a subspace?
Isn't the dimension just how many vectors there are? But I'm confused, because given W=span{(0, 2, 3) , (0, 1, -2), (1, 5,-7)}, I'd say the dimension is three, but it can't be that easy!

Same for V=span{(1, 0, -1), (2, 1, 1) , (0, 1, 3)}? How do I find the dimension for each of...

Answer 4

Sub spaces aren't necessarily finite, but quite often they have a finite dimension.

You need to find a basis for your subspace. The dimension is then the number of elements in this basis. Since all bases of a vector space have the same number of elements, this works nicely. Remember that a basis must be linearly independent.

The way you actually find a basis is to write your span in a matrix [in this case you would have a 3*3 matrix] and reduce it to echelon form. The basic columns of this echelon matrix indicate which columns in your original matrix form a basis. The number of basic columns is your dimension.

Answer 5

It is that easy [provided your given vectors are linearly independent].
And your vectors in W are linearly independent - two have a 0 x-value and the third doesn't so no combination of two of them can give the third. Dimension = 3.
For V = span{v_1, v_2, v_3,}
-2v_1 + v_2 = v_3. The third vector is just a result of combining the other 2 appropriately. v_1 nad v_2 are linearly independent, so dim = 2.

Answer 6

it is the minimum number of vectors that you need to span the space you want to span.

You find it by calculating basevetors for the space you are looking at.