Basis & Dimension: Subspace of R4

[ggb123]

Homework Statement

Find a basis and dimension to each of the following subspaces of R4:

U = {(a+b,a+c,b+c,a+b+c)|a,b,c∈R}

Homework Equations

The Attempt at a Solution

I started by making a linear system.
w(a + b) + x(a + c) + y(b + c) + z(a + b + c) = 0
a(w + x + z) + b(w + y + z) + c(x + y + z) = 0

w + x + 0 + z = 0
w + 0 + y + z = 0
0 + x + y + z = 0

Then I get stuck from here. Any help would be really appreciated, thanks.

 

[Mark44]

Homework Statement

Find a basis and dimension to each of the following subspaces of R4:

U = {(a+b,a+c,b+c,a+b+c)|a,b,c∈R}

Homework Equations

The Attempt at a Solution

I started by making a linear system.
w(a + b) + x(a + c) + y(b + c) + z(a + b + c) = 0
a(w + x + z) + b(w + y + z) + c(x + y + z) = 0

w + x + 0 + z = 0
w + 0 + y + z = 0
0 + x + y + z = 0

Then I get stuck from here. Any help would be really appreciated, thanks.

From the definition of set U,
x = a + b
y = a + ... + c
z = ... b + c
w = a + b + c

From this system of equations it can be seen that every vector u in U is a linear combination of three vectors.

 

[HallsofIvy]

A just slightly different way of writing the same thing:
(a+b,a+c,b+c,a+b+c)= (a, a, 0, a)+ (b, 0, b, b)+ (0, c, c, c)= a(1, 1, 0, 1)+ b(1, 0, 1, 1)+ c(0, 1, 1, 1). Those are the basis vectors.

In other words, don't introduce new constants, w, x,y z, take out the constants that are already there!