The dimension of the following subspaces

Question

The dimension of the following subspaces

1) all vectors of the form [a+c a−b b+c −a+b] of the vector space R4 is Answer here:   .

2) all vectors of the form [a b c d] of the vector space R4, where a=b is Answer here:  .

3) all vectors of the form [a b c d] of the vector space R4, where 2a=b is Answer here:  .

Answer 1

1. [a+c, a-b, b+c, -a+b] = a[1, 1, 0, -1] + b[0, -1, 1, 1] + c[1, 0, 1, 0]. You can reduce the 3x4 matrix with the rows [1, 1, 0, -1], [0, -1, 1, 1], and [1, 0, 1, 0] to row echelon form and see that the vectors are linearly dependent. Equivalently, we can see the linear dependence when a=b=1: 1[1, 1, 0, -1] + 1[0, -1, 1, 1]=[1, 0, 1, 0]. The vectors [1, 1, 0, -1] and [0, -1, 1, 1] are linearly independent, so [a+c, a-b, b+c, -a+b] spans the 2-dimensional subspace a[1, 1, 0, -1] + b[0, -1, 1, 1] = [a, a-b, b, -a+b] in R⁴.
2. [a, a, c, d] = a[1, 1, 0, 0] + c[0, 0, 1, 0] + d[0, 0, 0, 1]. The 3x4 matrix with the vectors [1, 1, 0, 0], [0, 0, 1, 0], and [0, 0, 0, 1] is already in row echelon form so the vectors are linearly independent. Therefore [a, a, c, d] spans a 3-dimensional subspace in R⁴.
3. [a, 2a, c, d] = a[1, 2, 0, 0] + c[0, 0, 1, 0] + d[0, 0, 0, 1]. The 3x4 matrix with the vectors [1, 2, 0, 0], [0, 0, 1, 0], and [0, 0, 0, 1] is already in row echelon form so the vectors are linearly independent. Therefore [a, 2a, c, d] spans a 3-dimensional subspace in R⁴.