What are the bases for U and W in R^4?

[abcdefg]

let V = R^{4}. Consider the following subspaces:

U = { (a,b,c,d) in R^{4} : a + 2b + 3c = 0, a + b + c + d = 0 }
W = { (a,b,c,d) in R^{4} : a + d = 0, b + c = 0 }

find a basis for U and a basis for W.

i don't even know where to begin. Any help would be very much appreciated. Thanks.

 

[amcavoy]

let V = R^{4}. Consider the following subspaces:

U = { (a,b,c,d) in R^{4} : a + 2b + 3c = 0, a + b + c + d = 0 }
W = { (a,b,c,d) in R^{4} : a + d = 0, b + c = 0 }

find a basis for U and a basis for W.

i don't even know where to begin. Any help would be very much appreciated. Thanks.

Look at the cooresponding matrix and reduce it to row-echelon form. The columns with pivots in them will be the same columns you use for your basis (remember, the same column... but from the ORIGINAL matrix).

 

[quicknote]

I can provide a mathematical response to this question. To find the bases for U and W in R^4, we need to determine the linearly independent vectors that span each subspace.

To find a basis for U, we can start by setting up a system of equations using the given conditions for U. We have:

a + 2b + 3c = 0
a + b + c + d = 0

We can rewrite these equations as:

a = -2b - 3c
d = -a - b - c

Substituting these values into the vector form, we get:

(a, b, c, d) = (-2b - 3c, b, c, -2b - 3c - b - c) = (-2b - 3c, b, c, -3b - 4c)

This means that any vector in U can be written as a linear combination of the vectors (-2, 1, 0, -3) and (-3, 0, 1, -4). These two vectors are linearly independent and span U, therefore they form a basis for U.

To find a basis for W, we can similarly set up a system of equations using the given conditions for W. We have:

a + d = 0
b + c = 0

We can rewrite these equations as:

a = -d
b = -c

Substituting these values into the vector form, we get:

(a, b, c, d) = (-d, -c, c, d)

This means that any vector in W can be written as a linear combination of the vectors (-1, 0, 1, 1) and (0, -1, 1, 0). These two vectors are linearly independent and span W, therefore they form a basis for W.

In summary, the basis for U is {(-2, 1, 0, -3), (-3, 0, 1, -4)} and the basis for W is {(-1, 0, 1, 1), (0, -1, 1, 0)}.